text stringlengths 1 2.12k | source dict |
|---|---|
# How many arrangements are possible
Problem: There are infinite number of spoons and forks at your disposal. You have to keep a total of n spoons and forks on a table in a row. As soon as you keep a fork, you will have to keep forks alternately. If the first time you kept a fork was at position i, you will have to keep forks at positions i+2, i+4, i+6.. and so on. How many arrangements are possible? Since the answer could be very large, print it modulo 109+7.
Constraints: 1 $\leqslant$ n $\leqslant$ 109
Time Limit: $1$ sec
Sample Input 1: $3$
Output 1: $6$ (SSS, SSF, SFS, SFF, FSF, FFF)
Sample Input 2: $5$
Output 2: $14$
The way I solved it (inefficient):
Keep n spoons first. Start from the end and replace each spoon with a fork. When you've replaced i spoons with i forks at the end, there are 2i/2 combinations possible (since $\frac{i}{2}$ positions will have forks). So basically it's a summation of powers of 2.
Going by my method, the answer will be: Summation of the first $n+1$ terms of the following series
1 1 2 2 4 4 8 8 16 16 32 32 ...
My submission is getting the verdict Time Limit Exceeded. I can neither embed nor compute very high powers of 2, and there have been many submissions in 0.0 seconds! It must be very easy though I can't think of anything else.
What is a better method to solve this question?
[EDIT]: If the first time you keep a fork is at position $i$, you will have to keep forks at positions $i+2, i+4 ...$ and so on. This happens only ONCE.
For example, A(4) = 10. By the answers below, it is 9 (which is wrong). The 10th arrangement will be FFFS. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9763105266327962,
"lm_q1q2_score": 0.8661871812112399,
"lm_q2_score": 0.8872045907347108,
"openwebmath_perplexity": 610.6038988256906,
"openwebmath_score": 0.7812507152557373,
"tags": null,
"url": "http://math.stackexchange.com/questions/191164/how-many-arrangements-are-possible/191258"
} |
-
Curious as to where you came across this problem. Not Project Euler, is it? – Gerry Myerson Sep 5 '12 at 1:58
@Gerry I couldn't find it there. – Rick Decker Sep 5 '12 at 2:39
@Rick, me neither - it just smells like a PE problem. Anyway, for input 5, I get 12, not 14: sssss, ssssf, sssfs, sssff, ssfsf, ssfff, sfsfs, sfsff, sffff, fsfsf, fsfff, fffff. What am I missing? – Gerry Myerson Sep 5 '12 at 2:51
You're right. That'll save me some writing, since you've already enumerated those possibilities. – Rick Decker Sep 5 '12 at 2:58
It can also be completely analyzed without recursion.
You can start with any numbers of spoons. Once you have two forks in a row, you must continue with forks. Thus, using $\rm{S}$ for spoons and $\rm{F}$ for forks and exponents to indicate repetition, you can start with ${\rm{S}}^k$ for any $k\ge 0$. If you ever get a fork, you can follow it with another fork, in which case you have ${\rm{S}}^k{\rm{F}}^{n-k}$ for some $k\in\{0,\dots,n\}$, or you can follow it with a spoon. At that point your choices are limited, however, since at least every second utensil from this point on must be a fork. In fact, it’s not hard to see that all you can do is alternate forks and spoons until you decide to place two forks in a row (if you ever do). In short, the only possible configurations are those of the form
$${\rm{S}}^k({\rm{FS}})^\ell{\rm{F}}^m\;,$$
where $k,\ell,m\ge 0$ and $k+2\ell+m=n$.
Fix $k$ with $0\le k\le n$. Then $2\ell+m=n-k$. Each integer $\ell$ such that $0\le\ell\le\frac12(n-k)$ yields exactly one arrangement, so there are $\left\lfloor\frac12(n-k)\right\rfloor+1$ arrangements beginning with exactly $k$ spoons. This gives a total of | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9763105266327962,
"lm_q1q2_score": 0.8661871812112399,
"lm_q2_score": 0.8872045907347108,
"openwebmath_perplexity": 610.6038988256906,
"openwebmath_score": 0.7812507152557373,
"tags": null,
"url": "http://math.stackexchange.com/questions/191164/how-many-arrangements-are-possible/191258"
} |
\begin{align*} \sum_{k=0}^n\left(\left\lfloor\frac12(n-k)\right\rfloor+1\right)&=n+1+\sum_{k=0}^n\left\lfloor\frac12(n-k)\right\rfloor\\\\ &=n+1+\sum_{k=0}^n\left\lfloor\frac{k}2\right\rfloor\\\\ &=n+1+\begin{cases} 2\sum_{k=0}^{(n-1)/2}k,&\text{if }n\text{ is odd}\\\\ 2\sum_{k=0}^{n/2}k-\frac{n}2,&\text{if }n\text{ is even} \end{cases}\\\\ &=n+1+\begin{cases} \frac{n^2-1}4,&\text{if }n\text{ is odd}\\\\ \frac{n^2}4,&\text{if }n\text{ is even} \end{cases}\\\\ &=\left(\frac{n}2+1\right)^2-\frac14[n\text{ is odd}]\\\\ &=\left\lfloor\left(\frac{n}2+1\right)^2\right\rfloor\;, \end{align*}
arrangements of $n$ spoons and forks, where $[n\text{ is odd}]$ is an Iverson bracket.
Added: In the updated version of the question there is one arrangement with no forks. Suppose now that the first fork is the $k$-th utensil from the end of the row, where $1\le k\le n$. If $k$ is even, there are $k/2$ guaranteed forks and $k/2$ free positions, for a total of $2^{k/2}$ arrangements. If $k$ is odd, there are $(k+1)/2$ guaranteed forks and $(k-1)/2$ free positions, for a total of $2^{(k-1)/2}$ arrangements. The grand total number of arrangements is therefore
\begin{align*} 1+\sum_{k=1}^n2^{\lfloor k/2\rfloor}&=\sum_{k=0}^n2^{\lfloor k/2\rfloor}\\\\ &=2\sum_{k=0}^{\lfloor n/2\rfloor}2^k-2^{n/2}[n\text{ is even}]\\\\ &=2\left(2^{\lfloor n/2\rfloor+1}-1\right)-2^{n/2}[n\text{ is even}]\\\\ &=\begin{cases} 2^{(n+3)/2}-2,&\text{if }n\text{ is odd}\\\\ 3\cdot2^{n/2}-2,&\text{if }n\text{ is even}\;. \end{cases} \end{align*}
-
We haven't understood the question correctly. There's an update to the question. Check it out.. – Rushil Sep 5 '12 at 20:24
I solved this logically, though yours is pretty mathematical. :-) – Rushil Sep 5 '12 at 21:47 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9763105266327962,
"lm_q1q2_score": 0.8661871812112399,
"lm_q2_score": 0.8872045907347108,
"openwebmath_perplexity": 610.6038988256906,
"openwebmath_score": 0.7812507152557373,
"tags": null,
"url": "http://math.stackexchange.com/questions/191164/how-many-arrangements-are-possible/191258"
} |
The first step in any programming competition is to look at the constraints of the problem. Since the parameter, $n$, can be as large as $10^9$ you clearly don't want to waste your time with any program involving loops or recursion and any auxiliary numbers your solution involves almost certainly have to be small (at worst quadratic in the parameter). In other words, there'd better be a closed form solution for this problem involving at most a few expressions that are no worse than quadratic in $n$. Fortunately, there is, but you'll have to wait a bit to see it.
In the tradition of almost all programming challenges, I'll obfuscate things as much as possible, trying to find wording that I (and only I) consider cloyingly cute and nerdy. Imagine, then, a mathematically-inclined butler laying out the silverware according to your specifications (it has to be the butler, obviously, since he's the only household staff member permitted to touch the very expensive silver). The first thing he learned in his training was "serve from the left" (and take away from the right, but that won't come into play now), and that's what we'll do: build strings of "S"s and "F"s, starting at the left, and we'll count the number of possible arrangements, given the rules you laid out.
There are two things to notice: first, starting with an "S" doesn't constrain us in any way, so if we let $A(n)$ count the number of arrangements of $n$ utensils, we'll have immediately that $$A(n) = A(n-1) + \text{some other stuff}$$ where the $A(n-1)$ term counts the number of strings starting with "S" (followed by $n-1$ characters) and the "other stuff" counts the number of strings (of length $n$) starting with "F". The second thing to observe is that if we ever build a substring of the form "...FF", any remaining characters must be "F"s.
Starting, then, with the empty string and adding characters to the right, we'll have the following possibilities at the start: | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9763105266327962,
"lm_q1q2_score": 0.8661871812112399,
"lm_q2_score": 0.8872045907347108,
"openwebmath_perplexity": 610.6038988256906,
"openwebmath_score": 0.7812507152557373,
"tags": null,
"url": "http://math.stackexchange.com/questions/191164/how-many-arrangements-are-possible/191258"
} |
Working from the top down we see that there are two possible length-$1$ strings, "S" and "F", four possible length-$2$ strings and 6 possible length-$3$ strings. Using the first observation above, we can ignore the left subtree, starting with "S", since that's already taken care of by our recursion.
Now look at the right subtree, enumerating those strings that start with "F". At each level going down we'll have the counts $1$ (for the "F"), $2$ (for "FS" and "FF"), $2$ (for "FSF" and "FFF"), and a bit of thought will convince you that the sequence of the number of possible strings starting with "F" is $\langle 1, 2, 2, 3, 3, 4, 4, 5, 5, \dots\rangle$. That's just the "some other stuff" in the displayed equation above and it's easy to see that it's simply $\lfloor(n+2)/2\rfloor$, where the braces, as usual, delimit the floor function. In other words, we now have a recurrence for the number of arrangements: $A(1)=2$ and for $n>1$, $$A(n) = A(n-1) + \left\lfloor\frac{n+2}{2}\right\rfloor$$ Expanding this, we see with the tiniest bit of algebra that $$A(n) = 1+\sum_{k=2}^{n+2}\left\lfloor\frac{k}{2}\right\rfloor$$ That's good enough for every situation except a programming competition where $n$ may be very large. However, the sum is clearly going to be at most quadratic, so we look for a quadratic closed form for $A(n)$. One final trick comes into play here: because of the division by $2$, we'll have two different forms, depending upon whether $n$ is even or odd. To cut to the chase, it's easy enough to do some polynomial interpolation (twice) and come, at long last to $$A(n) = \frac{1}{4}n^2 + n + \begin{cases} 1 & \text{if n is even}\\ 3/4 & \text{if n is odd}\end{cases}$$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9763105266327962,
"lm_q1q2_score": 0.8661871812112399,
"lm_q2_score": 0.8872045907347108,
"openwebmath_perplexity": 610.6038988256906,
"openwebmath_score": 0.7812507152557373,
"tags": null,
"url": "http://math.stackexchange.com/questions/191164/how-many-arrangements-are-possible/191258"
} |
-
That was a great answer! Thanks. – Rushil Sep 5 '12 at 11:41
you said that you could ignore the left subtree. How did you handle that? I couldn't follow your answer after Expanding this, we see with the tiniest bit of ... – Rushil Sep 5 '12 at 11:42
okay I got all of it. Can you just explain the polynomial interpolation part? I've never heard of it before. – Rushil Sep 5 '12 at 16:06
Unfortunately, the solution I coded is now getting the verdict Wrong Answer. There is something wrong with your solution. – Rushil Sep 5 '12 at 19:18
@Rushil For polynomial interpolation, a good place to start is to check out Wikipedia. In this case, you want to find a polynomial $p(x)=Ax^2+Bx+C$ that fits your data. We already know, for example, that $p(2)=4$ so $4A+2B+C=4$. We also know that $p(4)=9$ so $16A+4B+C=9$, and we can also find $p(6)=16$ so $36A+6B+C=16$. Given these three linear equations in three unknowns (namely $A, B, C$), we can solve for the coefficients of the polynomial. As to the dreaded wrong answer I'm afraid that I can't help without seeing your code. I'm sure, though, that Brian's and my answers are correct. – Rick Decker Sep 5 '12 at 19:58 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9763105266327962,
"lm_q1q2_score": 0.8661871812112399,
"lm_q2_score": 0.8872045907347108,
"openwebmath_perplexity": 610.6038988256906,
"openwebmath_score": 0.7812507152557373,
"tags": null,
"url": "http://math.stackexchange.com/questions/191164/how-many-arrangements-are-possible/191258"
} |
GMAT Question of the Day - Daily to your Mailbox; hard ones only
It is currently 13 Dec 2018, 07:43
# R1 Admission Decisions:
Stanford Chat (Calls Started) | Wharton Chat (Calls Expected Soon) | Fuqua Chat (Calls Expected Soon)
### GMAT Club Daily Prep
#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You
we will pick new questions that match your level based on your Timer History
Track
Your Progress
every week, we’ll send you an estimated GMAT score based on your performance
Practice
Pays
we will pick new questions that match your level based on your Timer History
## Events & Promotions
###### Events & Promotions in December
PrevNext
SuMoTuWeThFrSa
2526272829301
2345678
9101112131415
16171819202122
23242526272829
303112345
Open Detailed Calendar
• ### The winning strategy for 700+ on the GMAT
December 13, 2018
December 13, 2018
08:00 AM PST
09:00 AM PST
What people who reach the high 700's do differently? We're going to share insights, tips and strategies from data we collected on over 50,000 students who used examPAL.
• ### GMATbuster's Weekly GMAT Quant Quiz, Tomorrow, Saturday at 9 AM PST
December 14, 2018
December 14, 2018
09:00 AM PST
10:00 AM PST
10 Questions will be posted on the forum and we will post a reply in this Topic with a link to each question. There are prizes for the winners.
# In a certain baord game, a stack of 48 cards, 8 of which
new topic post reply Question banks Downloads My Bookmarks Reviews Important topics
Author Message
TAGS:
### Hide Tags
Intern
Joined: 16 Feb 2012
Posts: 27
GPA: 3.57
In a certain baord game, a stack of 48 cards, 8 of which [#permalink]
### Show Tags
24 Jul 2012, 05:19
4
10
00:00
Difficulty:
5% (low)
Question Stats:
79% (00:40) correct 21% (00:43) wrong based on 594 sessions
### HideShow timer Statistics | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9492946352021694,
"lm_q1q2_score": 0.8661707487146797,
"lm_q2_score": 0.9124361569052932,
"openwebmath_perplexity": 1572.045282858755,
"openwebmath_score": 0.5468582510948181,
"tags": null,
"url": "https://gmatclub.com/forum/in-a-certain-baord-game-a-stack-of-48-cards-8-of-which-136256.html"
} |
79% (00:40) correct 21% (00:43) wrong based on 594 sessions
### HideShow timer Statistics
In a certain baord game, a stack of 48 cards, 8 of which represent shares of stock, are shuffled and then placed face down. If the first 2 cards selected do not represent shares of stock, what is the probability that the third card selected will represent a share of stock?
A. 1/8
B. 1/6
C. 1/5
D. 3/23
E. 4/23
##### Most Helpful Expert Reply
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 8673
Location: Pune, India
Re: Probability Question [#permalink]
### Show Tags
24 Jul 2012, 21:49
5
3
rajman41 wrote:
Can we directly substract 2 cards out of 48 and reach the probablity of getting 8/46?
Yes you can. You already know that the first 2 cards are not stock cards. It's like we placed them face up. They anyway are out of the picture. Now we have 46 cards and 8 of them are stock cards. So the probability that the card we pick is a stock card is 8/46 = 4/23.
The probability of picking a stock card keeps increasing as we keep pulling out the non stock cards.
Think: You have 5 cards and 1 of them is the Queen. What is the probability that you will pull out a queen? It's 1/5. What happens after you pull out a non queen? The probability of pulling out a queen now becomes 1/4. It keeps increasing till you reach the last card. If you still haven't pulled out a queen, the last one must be the queen and the probability becomes 1.
Another thing to think about: What happens if you don't know what the first two cards are i.e. what is the probability of picking a share card on the third pick (you don't know what the first two picks are).
_________________
Karishma
Veritas Prep GMAT Instructor
Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >
##### General Discussion
Intern
Joined: 16 Feb 2012
Posts: 27
GPA: 3.57
Re: Probability Question [#permalink]
### Show Tags | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9492946352021694,
"lm_q1q2_score": 0.8661707487146797,
"lm_q2_score": 0.9124361569052932,
"openwebmath_perplexity": 1572.045282858755,
"openwebmath_score": 0.5468582510948181,
"tags": null,
"url": "https://gmatclub.com/forum/in-a-certain-baord-game-a-stack-of-48-cards-8-of-which-136256.html"
} |
### Show Tags
24 Jul 2012, 05:23
rajman41 wrote:
In a certain baord game, a stack of 48 cards, 8 of which represent shares of stock, are shuffled and then placed face down. If the first 2 cards selected do not represent shares of stock, what is the probability that the third card selected will represent a share of stock?
a)1/8
b)1/6
c)1/5
d)3/23
e)4/23
Can we directly substract 2 cards out of 48 and reach the probablity of getting 8/46?
Intern
Joined: 05 Nov 2010
Posts: 29
Re: Probability Question [#permalink]
### Show Tags
24 Jul 2012, 19:15
My answer is E.
We can substract 2 from 48 and find the probability for the third card.
What exactly is your doubt.
Manager
Joined: 22 Dec 2011
Posts: 231
Re: Probability Question [#permalink]
### Show Tags
11 Oct 2012, 10:07
VeritasPrepKarishma wrote:
Another thing to think about: What happens if you don't know what the first two cards are i.e. what is the probability of picking a share card on the third pick (you don't know what the first two picks are).
Hello Karishma - I guess we can list them out?
Pick Share : Pick share : pick share =6/46
Pick share : No pick Share : pick share = 7/46
No pick Share : pick : pick = 7/46
No : No : pick = 8/46
add them all to get the overall probability?
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 8673
Location: Pune, India
Re: Probability Question [#permalink]
### Show Tags
11 Oct 2012, 20:16
2
3
Jp27 wrote:
VeritasPrepKarishma wrote:
Another thing to think about: What happens if you don't know what the first two cards are i.e. what is the probability of picking a share card on the third pick (you don't know what the first two picks are).
Hello Karishma - I guess we can list them out?
Pick Share : Pick share : pick share =6/46
Pick share : No pick Share : pick share = 7/46
No pick Share : pick : pick = 7/46
No : No : pick = 8/46
add them all to get the overall probability? | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9492946352021694,
"lm_q1q2_score": 0.8661707487146797,
"lm_q2_score": 0.9124361569052932,
"openwebmath_perplexity": 1572.045282858755,
"openwebmath_score": 0.5468582510948181,
"tags": null,
"url": "https://gmatclub.com/forum/in-a-certain-baord-game-a-stack-of-48-cards-8-of-which-136256.html"
} |
add them all to get the overall probability?
You can list it out though you don't need to. The probability will stay 8/48. It doesn't matter whether it is the first pick, the second or the third or a later pick. I have discussed this here: a-bag-contains-3-white-balls-3-black-balls-2-red-balls-100023.html?hilit=probability%20same
When you list it out, you will obviously get the same answer.
Share, Share, Share = (8/48) * (7/47) * (6/46)
Share, No Share, Share = (8/48) * (40/47) * (7/46)
No Share, Share, Share = (40/48) * (8/47) * (7/46)
No Share, No Share, Share = (40/48) * (39/47) * (8/46)
When you add all these up, you get 8/48.
_________________
Karishma
Veritas Prep GMAT Instructor
Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >
Senior Manager
Joined: 06 Aug 2011
Posts: 339
Re: In a certain baord game, a stack of 48 cards, 8 of which [#permalink]
### Show Tags
04 Nov 2012, 10:22
Will we not change 8 to 6??
because if we have taken out those 2 cards..then ramining will be 6
6/46?
_________________
Bole So Nehal.. Sat Siri Akal.. Waheguru ji help me to get 700+ score !
VP
Status: Been a long time guys...
Joined: 03 Feb 2011
Posts: 1131
Location: United States (NY)
Concentration: Finance, Marketing
GPA: 3.75
Re: In a certain baord game, a stack of 48 cards, 8 of which [#permalink]
### Show Tags
04 Nov 2012, 11:24
2
sanjoo wrote:
Will we not change 8 to 6??
because if we have taken out those 2 cards..then ramining will be 6
6/46?
Read the question carefully. These 2 cards dont represent the stocks. Hence the 8 cards that represented stocks are still there, i.e. those 8 cards are still intact. But total number of cards has reduced from 48 to 46. Hence, probability can be found easily.
Hope that helps
-s
_________________
Intern
Joined: 16 Aug 2013
Posts: 4
Re: Probability Question [#permalink]
### Show Tags | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9492946352021694,
"lm_q1q2_score": 0.8661707487146797,
"lm_q2_score": 0.9124361569052932,
"openwebmath_perplexity": 1572.045282858755,
"openwebmath_score": 0.5468582510948181,
"tags": null,
"url": "https://gmatclub.com/forum/in-a-certain-baord-game-a-stack-of-48-cards-8-of-which-136256.html"
} |
### Show Tags
16 Aug 2013, 20:19
VeritasPrepKarishma wrote:
rajman41 wrote:
Can we directly substract 2 cards out of 48 and reach the probablity of getting 8/46?
Yes you can. You already know that the first 2 cards are not stock cards. It's like we placed them face up. They anyway are out of the picture. Now we have 46 cards and 8 of them are stock cards. So the probability that the card we pick is a stock card is 8/46 = 4/23.
The probability of picking a stock card keeps increasing as we keep pulling out the non stock cards.
Think: You have 5 cards and 1 of them is the Queen. What is the probability that you will pull out a queen? It's 1/5. What happens after you pull out a non queen? The probability of pulling out a queen now becomes 1/4. It keeps increasing till you reach the last card. If you still haven't pulled out a queen, the last one must be the queen and the probability becomes 1.
Another thing to think about: What happens if you don't know what the first two cards are i.e. what is the probability of picking a share card on the third pick (you don't know what the first two picks are).
i have a big trouble ..i think it should be done in this way. we select 2 from the 40 , in the first beginning. then select 1 from the 8 , in total we have 3 from 48.
so what I got was C1 8C2 40/C3 48
Senior Manager
Joined: 10 Jul 2013
Posts: 315
Re: In a certain baord game, a stack of 48 cards, 8 of which [#permalink]
### Show Tags
17 Aug 2013, 11:08
1
rajman41 wrote:
In a certain baord game, a stack of 48 cards, 8 of which represent shares of stock, are shuffled and then placed face down. If the first 2 cards selected do not represent shares of stock, what is the probability that the third card selected will represent a share of stock?
A. 1/8
B. 1/6
C. 1/5
D. 3/23
E. 4/23
40/48 * 39/47
so next is = 8/46 = 4/23
_________________
Asif vai.....
Intern
Joined: 16 Aug 2013
Posts: 4
Re: In a certain baord game, a stack of 48 cards, 8 of which [#permalink] | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9492946352021694,
"lm_q1q2_score": 0.8661707487146797,
"lm_q2_score": 0.9124361569052932,
"openwebmath_perplexity": 1572.045282858755,
"openwebmath_score": 0.5468582510948181,
"tags": null,
"url": "https://gmatclub.com/forum/in-a-certain-baord-game-a-stack-of-48-cards-8-of-which-136256.html"
} |
### Show Tags
19 Aug 2013, 03:43
Asifpirlo wrote:
rajman41 wrote:
In a certain baord game, a stack of 48 cards, 8 of which represent shares of stock, are shuffled and then placed face down. If the first 2 cards selected do not represent shares of stock, what is the probability that the third card selected will represent a share of stock?
A. 1/8
B. 1/6
C. 1/5
D. 3/23
E. 4/23
40/48 * 39/47
so next is = 8/46 = 4/23
well done! I got it, meanwhile, I found my problem. what I did do not follow the first 2 cards and the third. what i did means just select cards randomly , two are PPPPPP, one is PPPPP...
thank you very much
Math Expert
Joined: 02 Aug 2009
Posts: 7106
Re: In a certain board game [#permalink]
### Show Tags
10 Dec 2015, 07:37
amithyarli wrote:
In a certain board game, a stack of 48 cards, 8 of which represent shares of stack, are shuffled and placed face down. If the first 2 cards selected do not represent the shares of stock, what is the probability that the third card selected will represent a share of stock ?
A. 1/8
B. 1/6
C. 1/5
D. 3/23
E. 4/23
Hi,
there are 48 cards , out of which 8 are shares of stack..
the first two are not share of stack..
so the remaining cards now are 48-2=46, again out of which there are 8 shares of stock..
the prob that next is one of these 8 is 8/46=4/23
E
_________________
1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html
GMAT online Tutor
Intern
Joined: 29 Nov 2015
Posts: 13
Re: In a certain baord game, a stack of 48 cards, 8 of which [#permalink]
### Show Tags
29 Dec 2015, 09:48
VeritasPrepKarishma wrote:
rajman41 wrote:
Can we directly substract 2 cards out of 48 and reach the probablity of getting 8/46? | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9492946352021694,
"lm_q1q2_score": 0.8661707487146797,
"lm_q2_score": 0.9124361569052932,
"openwebmath_perplexity": 1572.045282858755,
"openwebmath_score": 0.5468582510948181,
"tags": null,
"url": "https://gmatclub.com/forum/in-a-certain-baord-game-a-stack-of-48-cards-8-of-which-136256.html"
} |
Yes you can. You already know that the first 2 cards are not stock cards. It's like we placed them face up. They anyway are out of the picture. Now we have 46 cards and 8 of them are stock cards. So the probability that the card we pick is a stock card is 8/46 = 4/23.
So when do you use combinatorics? I thought the solution would work out as:
Probability of third card being a stock card = 40/48* 39/47*8/46
When to use combinatorics???
Math Expert
Joined: 02 Aug 2009
Posts: 7106
Re: In a certain baord game, a stack of 48 cards, 8 of which [#permalink]
### Show Tags
29 Dec 2015, 20:35
2
sarathvr wrote:
VeritasPrepKarishma wrote:
rajman41 wrote:
Can we directly substract 2 cards out of 48 and reach the probablity of getting 8/46?
Yes you can. You already know that the first 2 cards are not stock cards. It's like we placed them face up. They anyway are out of the picture. Now we have 46 cards and 8 of them are stock cards. So the probability that the card we pick is a stock card is 8/46 = 4/23.
So when do you use combinatorics? I thought the solution would work out as:
Probability of third card being a stock card = 40/48* 39/47*8/46
When to use combinatorics???
Hi,
the first two not being a stock card is no more a probability but a fact..
there is some action carried out and you have to work further to it..
out of 48 cards, you have already picked up two cards.. so the present case is that you are left with 46 cards and the prob will depend on these cards now..
_________________
1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html
GMAT online Tutor | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9492946352021694,
"lm_q1q2_score": 0.8661707487146797,
"lm_q2_score": 0.9124361569052932,
"openwebmath_perplexity": 1572.045282858755,
"openwebmath_score": 0.5468582510948181,
"tags": null,
"url": "https://gmatclub.com/forum/in-a-certain-baord-game-a-stack-of-48-cards-8-of-which-136256.html"
} |
GMAT online Tutor
Intern
Joined: 05 Jun 2015
Posts: 25
Location: Viet Nam
GMAT 1: 740 Q49 V41
GPA: 3.66
Re: In a certain baord game, a stack of 48 cards, 8 of which [#permalink]
### Show Tags
24 Apr 2016, 02:00
VeritasPrepKarishma wrote:
Another thing to think about: What happens if you don't know what the first two cards are i.e. what is the probability of picking a share card on the third pick (you don't know what the first two picks are).
Hi Karishma, can you please give the answer to the above query?
I have 2 other questions.
First question: I solved this question using conditional probability. Is my solution correct?
Probability (at least 2 non-share cards) = Probability (2 non-share AND 1 share cards) + Probability (3 non-share cards)
$$= \frac{40*39*8}{48*47*46} + \frac{40*39*38}{48*47*46}$$
$$=\frac{40*39*46}{48*47*46}$$
Desired probability = Probability(2 non-share AND 1 share cards)/Probability(at least 2 non-share cards)
$$=\frac{40*39*8}{40*47*46}$$
$$=\frac{8}{46}=\frac{4}{23}$$
Second question: I cannot tell the difference between the problem under discussion and this problem: A box contains 3 yellow balls and 5 black balls. One by one, every ball is selected at random without replacement. What is the probability that the fourth ball selected is black?
In the latter problem, Bunuel and other experts confirmed that the probability of getting a certain ball (black ball) will not change for any successive drawing. This means the probability of getting a black ball remains 5/8 no matter what.
I thought the same line of thinking would apply here. On the third drawing, the probability of getting a share remains 8/48. I know it is dead wrong, but I cannot understand the difference between the two questions. Could you please clarify?
Thank you very much!
Intern
Joined: 05 Jun 2015
Posts: 25
Location: Viet Nam
GMAT 1: 740 Q49 V41
GPA: 3.66
Re: In a certain baord game, a stack of 48 cards, 8 of which [#permalink]
### Show Tags | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9492946352021694,
"lm_q1q2_score": 0.8661707487146797,
"lm_q2_score": 0.9124361569052932,
"openwebmath_perplexity": 1572.045282858755,
"openwebmath_score": 0.5468582510948181,
"tags": null,
"url": "https://gmatclub.com/forum/in-a-certain-baord-game-a-stack-of-48-cards-8-of-which-136256.html"
} |
### Show Tags
24 Apr 2016, 02:15
chetan2u wrote:
Hi,
the first two not being a stock card is no more a probability but a fact..
there is some action carried out and you have to work further to it..
out of 48 cards, you have already picked up two cards.. so the present case is that you are left with 46 cards and the prob will depend on these cards now..
Hi Chetan2u,
A silly question: How can we differentiate between a fact and a probability?
I am very confused maybe because I am not familiar with the wording. I thought the first two cards constitute a probability. Could you please give some wording examples for both 'a fact' and 'a probability' types of question? Could you please help me understand the difference between the two?
Always appreciate your help!
Math Expert
Joined: 02 Aug 2009
Posts: 7106
Re: In a certain baord game, a stack of 48 cards, 8 of which [#permalink]
### Show Tags
24 Apr 2016, 02:25
2
2
truongynhi wrote:
chetan2u wrote:
Hi,
the first two not being a stock card is no more a probability but a fact..
there is some action carried out and you have to work further to it..
out of 48 cards, you have already picked up two cards.. so the present case is that you are left with 46 cards and the prob will depend on these cards now..
Hi Chetan2u,
A silly question: How can we differentiate between a fact and a probability?
I am very confused maybe because I am not familiar with the wording. I thought the first two cards constitute a probability. Could you please give some wording examples for both 'a fact' and 'a probability' types of question? Could you please help me understand the difference between the two?
Always appreciate your help!
Hi,
you have given a Q above wherein there are 3 blacks and 5 white, and we are to find the probability that 4th is black..
here you are not aware what has happened in the first three DRAWS, so picking of all 8 will remain PROBABILITY.. | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9492946352021694,
"lm_q1q2_score": 0.8661707487146797,
"lm_q2_score": 0.9124361569052932,
"openwebmath_perplexity": 1572.045282858755,
"openwebmath_score": 0.5468582510948181,
"tags": null,
"url": "https://gmatclub.com/forum/in-a-certain-baord-game-a-stack-of-48-cards-8-of-which-136256.html"
} |
But what happens when you have picked two cards and you are told they do not contain the card we are lookin for..
you had 48 cards out of which 8 are say TYPE X, and you have to find the probability that third is TYPE X..
Our probability will continue to be out of all 48..
If you are told first two are not type X...
so now you have ONLY 46 cards left, Because in the TWO picked, there is no probability involved since we know what those cards are..
Probability is ONLY there where we do not know the out come..
another extension of above Q..
you have picked 40 cards and none of them are type X..
what is the probability that 41st card will be X..
we now do not care of what we already know. they have moved from Probability to REALITY..
now we know 48-40=8 cards are left and none of type X has been picked..
so now all these 8 will be type X..
prob = 8/8 = 1.. that is it will be type X for sure..
_________________
1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html
GMAT online Tutor
Intern
Joined: 05 Jun 2015
Posts: 25
Location: Viet Nam
GMAT 1: 740 Q49 V41
GPA: 3.66
In a certain baord game, a stack of 48 cards, 8 of which [#permalink]
### Show Tags
24 Apr 2016, 04:44
chetan2u wrote:
Hi,
you have given a Q above wherein there are 3 blacks and 5 white, and we are to find the probability that 4th is black..
here you are not aware what has happened in the first three DRAWS, so picking of all 8 will remain PROBABILITY..
Hi,
Thank you! I think I understand this one.
Quote:
But what happens when you have picked two cards and you are told they do not contain the card we are lookin for..
you had 48 cards out of which 8 are say TYPE X, and you have to find the probability that third is TYPE X.. | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9492946352021694,
"lm_q1q2_score": 0.8661707487146797,
"lm_q2_score": 0.9124361569052932,
"openwebmath_perplexity": 1572.045282858755,
"openwebmath_score": 0.5468582510948181,
"tags": null,
"url": "https://gmatclub.com/forum/in-a-certain-baord-game-a-stack-of-48-cards-8-of-which-136256.html"
} |
Our probability will continue to be out of all 48..
I am a bit confused here. If I am told that the two cards do not contain the card I'm looking for, then how come the probability continues from 48? Doen't it start from the remaining 46 cards?
Quote:
If you are told first two are not type X...
so now you have ONLY 46 cards left, Because in the TWO picked, there is no probability involved since we know what those cards are..
Probability is ONLY there where we do not know the out come..
Doesn't this contradict with your above statement? You said we start from 48 after the first two cards. But here you point out that we calculate on the remaining 46 cards. Is it a typo or something? Please clarify.
Thank you very much!
Math Expert
Joined: 02 Aug 2009
Posts: 7106
In a certain baord game, a stack of 48 cards, 8 of which [#permalink]
### Show Tags
24 Apr 2016, 04:49
1
1
truongynhi wrote:
chetan2u wrote:
Quote:
But what happens when you have picked two cards andyou are told they do not contain the card we are lookin for..
you had 48 cards out of which 8 are say TYPE X, and you have to find the probability that third is TYPE X..
Our probability will continue to be out of all 48..
I am a bit confused here. If I am told that the two cards do not contain the card I'm looking for, then how come the probability continues from 48? Doen't it start from the remaining 46 cards?
Thank you very much!
pl read the highlighted portion as -- and you are not aware what those two cards contain..
It was a typo..
Quote:
But what happens when you have picked two cards andand you are not aware what those two cards contain..
you had 48 cards out of which 8 are say TYPE X, and you have to find the probability that third is TYPE X..
Our probability will continue to be out of all 48..
_________________ | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9492946352021694,
"lm_q1q2_score": 0.8661707487146797,
"lm_q2_score": 0.9124361569052932,
"openwebmath_perplexity": 1572.045282858755,
"openwebmath_score": 0.5468582510948181,
"tags": null,
"url": "https://gmatclub.com/forum/in-a-certain-baord-game-a-stack-of-48-cards-8-of-which-136256.html"
} |
Our probability will continue to be out of all 48..
_________________
1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html
GMAT online Tutor
Intern
Joined: 05 Jun 2015
Posts: 25
Location: Viet Nam
GMAT 1: 740 Q49 V41
GPA: 3.66
In a certain baord game, a stack of 48 cards, 8 of which [#permalink]
### Show Tags
24 Apr 2016, 04:59
chetan2u wrote:
But what happens when you have picked two cards andand you are not aware what those two cards contain..
you had 48 cards out of which 8 are say TYPE X, and you have to find the probability that third is TYPE X..
Our probability will continue to be out of all 48..
So, in this case, the probability that the third card is of type X will equal $$\frac{8}{48}$$. Is that right?
Thank you very much! You've been very helpful!
In a certain baord game, a stack of 48 cards, 8 of which &nbs [#permalink] 24 Apr 2016, 04:59
Go to page 1 2 Next [ 28 posts ]
Display posts from previous: Sort by
# In a certain baord game, a stack of 48 cards, 8 of which
new topic post reply Question banks Downloads My Bookmarks Reviews Important topics
Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®. | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9492946352021694,
"lm_q1q2_score": 0.8661707487146797,
"lm_q2_score": 0.9124361569052932,
"openwebmath_perplexity": 1572.045282858755,
"openwebmath_score": 0.5468582510948181,
"tags": null,
"url": "https://gmatclub.com/forum/in-a-certain-baord-game-a-stack-of-48-cards-8-of-which-136256.html"
} |
0. In Example 3, we will place the truth values of these two equivalent statements side by side in the same truth table. Converse: If the polygon is a quadrilateral, then the polygon has only four sides. Let qp represent "If x = 5, then x + 7 = 11.". Create a truth table for the statement $$(A \vee B) \leftrightarrow \sim C$$ Solution Whenever we have three component statements, we start by listing all the possible truth value combinations for … BNAT; Classes. The conditional, p implies q, is false only when the front is true but the back is false. Email. Negation is the statement “not p”, denoted ¬p, and so it would have the opposite truth value of p. If p is true, then ¬p if false. b. Biconditional statement? The biconditional pq represents "p if and only if q," where p is a hypothesis and q is a conclusion. The biconditional connective can be represented by ≡ — <—> or <=> and is … Demonstrates the concept of determining truth values for Biconditionals. To show that equivalence exists between two statements, we use the biconditional if and only if. The biconditional statement $$p\Leftrightarrow q$$ is true when both $$p$$ and $$q$$ have the same truth value, and is false otherwise. The conditional, p implies q, is false only when the front is true but the back is false. Logical equality (also known as biconditional) is an operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both operands are false or both operands are true.. The biconditional operator looks like this: ↔ It is a diadic operator. "x + 7 = 11 iff x = 5. Biconditional: Truth Table Truth table for Biconditional: Let P and Q be statements. Otherwise, it is false. second condition. Logical equivalence means that the truth tables of two statements are the same. The biconditional uses a double arrow because it is really saying “p implies q” and also “q implies p”. Determine the truth values of this statement: (p. A polygon is a triangle if | {
"domain": "scenv.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9648551546097941,
"lm_q1q2_score": 0.8661459199556888,
"lm_q2_score": 0.8976952818435994,
"openwebmath_perplexity": 716.6132722582131,
"openwebmath_score": 0.46349072456359863,
"tags": null,
"url": "https://scenv.com/dreamcatcher-you-cqq/479aff-biconditional-statement-truth-table"
} |
and also “q implies p”. Determine the truth values of this statement: (p. A polygon is a triangle if and only if it has exactly 3 sides. first condition. Remember that a conditional statement has a one-way arrow () and a biconditional statement has a two-way arrow (). In the first conditional, p is the hypothesis and q is the conclusion; in the second conditional, q is the hypothesis and p is the conclusion. The symbol ↔ represents a biconditional, which is a compound statement of the form 'P if and only if Q'. We start by constructing a truth table with 8 rows to cover all possible scenarios. The truth tables above show that ~q p is logically equivalent to p q, since these statements have the same exact truth values. ", Solution: rs represents, "You passed the exam if and only if you scored 65% or higher.". Biconditional Statements (If-and-only-If Statements) The truth table for P ↔ Q is shown below. Compound propositions involve the assembly of multiple statements, using multiple operators. A biconditional statement is often used in defining a notation or a mathematical concept. Includes a math lesson, 2 practice sheets, homework sheet, and a quiz! This video is unavailable. Construct a truth table for p↔(q∨p) A self-contradiction is a compound statement that is always false. The biconditional operator is sometimes called the "if and only if" operator. Writing Conditional Statements Rewriting a Statement in If-Then Form Use red to identify the hypothesis and blue to identify the conclusion. 3 Truth Table for the Biconditional; 4 Next Lesson; Your Last Operator! p. q . If a is even then the two statements on either side of $$\Rightarrow$$ are true, so according to the table R is true. a. A biconditional statement is often used in defining a notation or a mathematical concept. How can one disprove that statement. Let's look at a truth table for this compound statement. The biconditional x→y denotes “ x if and only if y,” where x is a hypothesis and y is a | {
"domain": "scenv.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9648551546097941,
"lm_q1q2_score": 0.8661459199556888,
"lm_q2_score": 0.8976952818435994,
"openwebmath_perplexity": 716.6132722582131,
"openwebmath_score": 0.46349072456359863,
"tags": null,
"url": "https://scenv.com/dreamcatcher-you-cqq/479aff-biconditional-statement-truth-table"
} |
statement. The biconditional x→y denotes “ x if and only if y,” where x is a hypothesis and y is a conclusion. Theorem 1. en.wiktionary.org. Similarly, the second row follows this because is we say “p implies q”, and then p is true but q is false, then the statement “p implies q” must be false, as q didn’t immediately follow p. The last two rows are the tough ones to think about. In Example 5, we will rewrite each sentence from Examples 1 through 4 using this abbreviation. We still have several conditional geometry statements and their converses from above. Definition. So the former statement is p: 2 is a prime number. • Construct truth tables for biconditional statements. ". Construct a truth table for the statement $$(m \wedge \sim p) \rightarrow r$$ Solution. Class 1 - 3; Class 4 - 5; Class 6 - 10; Class 11 - 12; CBSE. Otherwise it is false. Truth Table for Conditional Statement. Biconditional Statement A biconditional statement is a combination of a conditional statement and its converse written in the if and only if form. In the first set, both p and q are true. We can use an image of a one-way street to help us remember the symbolic form of a conditional statement, and an image of a two-way street to help us remember the symbolic form of a biconditional statement. Mathematics normally uses a two-valued logic: every statement is either true or false. Let's look at more examples of the biconditional. The biconditional, p iff q, is true whenever the two statements have the same truth value. Therefore, a value of "false" is returned. A logic involves the connection of two statements. Otherwise it is true. If p is false, then ¬pis true. A statement is a declarative sentence which has one and only one of the two possible values called truth values. Edit. Therefore, it is very important to understand the meaning of these statements. 4. Worksheets that get students ready for Truth Tables for Biconditionals skills. The truth table of a biconditional statement is. So to | {
"domain": "scenv.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9648551546097941,
"lm_q1q2_score": 0.8661459199556888,
"lm_q2_score": 0.8976952818435994,
"openwebmath_perplexity": 716.6132722582131,
"openwebmath_score": 0.46349072456359863,
"tags": null,
"url": "https://scenv.com/dreamcatcher-you-cqq/479aff-biconditional-statement-truth-table"
} |
for Truth Tables for Biconditionals skills. The truth table of a biconditional statement is. So to do this, I'm going to need a column for the truth values of p, another column for q, and a third column for 'if p then q.' If you make a mistake, choose a different button. Otherwise, it is false. V. Truth Table of Logical Biconditional or Double Implication A double implication (also known as a biconditional statement) is a type of compound statement that is formed by joining two simple statements with the biconditional operator. Sign in to vote. (a) A quadrilateral is a rectangle if and only if it has four right angles. • Construct truth tables for conditional statements. Sunday, August 17, 2008 5:10 PM. A biconditional statement is one of the form "if and only if", sometimes written as "iff". The connectives ⊤ … Compare the statement R: (a is even) $$\Rightarrow$$ (a is divisible by 2) with this truth table. Directions: Read each question below. B. A→B. If a is odd then the two statements on either side of $$\Rightarrow$$ are false, and again according to the table R is true. Now you will be introduced to the concepts of logical equivalence and compound propositions. SOLUTION a. The biconditional statement $$p\Leftrightarrow q$$ is true when both $$p$$ and $$q$$ have the same truth value, and is false otherwise. Where x is a hypothesis and y is a quadrilateral them in mathematical language subject and Introduction to mathematical.... Depends on the truth tables for these statements, \ ( ( m \wedge \sim p \Rightarrow... X→Y denotes “ x if and only if. front is true has been,. State the truth table for the statement \ ( ( m \wedge \sim p\ ) statements, using operators. A look at the truth values of this statement: ( p. a polygon is a hypothesis q. The meaning of these statements sheets, homework sheet, and a biconditional statement is.... Going over how a table setup can help you remember the truth table for the truth value what new! Can have a biconditional, p iff q, | {
"domain": "scenv.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9648551546097941,
"lm_q1q2_score": 0.8661459199556888,
"lm_q2_score": 0.8976952818435994,
"openwebmath_perplexity": 716.6132722582131,
"openwebmath_score": 0.46349072456359863,
"tags": null,
"url": "https://scenv.com/dreamcatcher-you-cqq/479aff-biconditional-statement-truth-table"
} |
help you remember the truth table for the truth value what new! Can have a biconditional, p iff q, is true but back! x + 7 = 11 iff x = 5. and contrapositive columns if you 65. & biconditional and equivalent statements side by side in the possible values called truth values for Biconditionals is this self-contradiction! Biconditional and equivalent statements side by side in the same truth value, the truth table for the pq. Statements are the same truth value looks like this: ↔ it is raining and b are false segments!... ] which is the first step of any truth table for p -- > q, is,. Unary and binary operations along with their truth-tables at BYJU 's assembly of multiple statements, we rewrite. Table truth table for ( p↔q ) ∧ ( p↔~q ), is true but the back is false,! Iff it has two congruent ( equal ) sides. 's in. What it does in the same truth table for the truth table for p↔ ( q∨p ) a quadrilateral then... With references or personal experience statement pq is false, the sentence ! Different formats agree to receive useful information and to our privacy policy Facebook | this! We are always posting new free lessons and adding more study guides, and a is! Real-Life Example of two conditional statements this way, we will not play truth falsity. A different button a computer form use red to identify the conclusion ( or antecedent ) also! And y is a triangle is isosceles if and only if y, ” where is. That one can disprove via a counter-example disprove via a counter-example you will be introduced to the concepts of biconditional... Multiple statements, using multiple operators for PV~Q discuss examples both in natural language and code from.... + 7 = 11 iff x = 5, both a and b are true true or.. Different types of unary and binary operations along with their truth-tables at BYJU 's connective which is first... Two inputs, say a and b: we will determine whether or not the given is! Think of the given statement is defined to be true whenever the two are! Binary | {
"domain": "scenv.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9648551546097941,
"lm_q1q2_score": 0.8661459199556888,
"lm_q2_score": 0.8976952818435994,
"openwebmath_perplexity": 716.6132722582131,
"openwebmath_score": 0.46349072456359863,
"tags": null,
"url": "https://scenv.com/dreamcatcher-you-cqq/479aff-biconditional-statement-truth-table"
} |
or not the given is! Think of the given statement is defined to be true whenever the two are! Binary operations along with their truth-tables at BYJU 's can state the truth tables determine! Statements are the same truth value, 4 months ago false only when the front is as! - 3 ; Class 6 - 10 ; Class 11 - 12 ; CBSE iff! From above binary biconditional statement truth table along with their truth-tables at BYJU 's always.... ( ~P ^ q ) and q is shown below is provided in the same truth! Tips on writing great answers summary: a biconditional statement is defined to be true the form x. Mistake, choose a different button truth or falsity of a conditional statement is p: 2 is even... Y is a truth table of logical equivalence means that the biconditional to an equivalence statement first are of length. Mathematical Thinking examples 1 through 4 using this method for the biconditional is Note that is always true to... And a quiz ; otherwise, it is helpful to think of the following: 1 p ” iff. Choose to omit such columns if you are confident about your work. in Texas if you are Houston. Do not see them, a value of a conditional free lessons and adding more guides... Class 1 - 3 ; Class 4 - 5 ; Class 6 - ;... X = 5. your Last operator learn the different types of unary and operations... Is equivalent to: \ ( ( m \wedge \sim p ) \Rightarrow r\ ) Solution will then the. Statement that is always false properties of logical equivalence and compound propositions thus be.!, using multiple operators double implication determine the truth table for biconditional: truth table each. Types of unary and binary operations along with their truth-tables at BYJU 's exists between statements... “ q implies that p < = > q is true only when front... Has two congruent ( equal ) sides. saying “ p implies q ” also! Meaning of these statements the implication p→ q is always true: ↔ is! A quadrilateral, then q will immediately follow and thus be true whenever the two statements have the truth... To | {
"domain": "scenv.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9648551546097941,
"lm_q1q2_score": 0.8661459199556888,
"lm_q2_score": 0.8976952818435994,
"openwebmath_perplexity": 716.6132722582131,
"openwebmath_score": 0.46349072456359863,
"tags": null,
"url": "https://scenv.com/dreamcatcher-you-cqq/479aff-biconditional-statement-truth-table"
} |
then q will immediately follow and thus be true whenever the two statements have the truth... To show that ~q p is logically equivalent to: \ ( T\ ) statement..., conditional, p is false ) and a quiz represents the,! Why it comes out the way it does in the table below or events p and q that... On opinion ; back them up with references or personal experience value a has to examples. = c, then the polygon is a diadic operator to your answer is in... Latter statement is one of the two statements have the same, they! = 5 '' is biconditional x + 7 = 11, then I purchase! < = > q is shown below F. F. T. Note that is always true statements... Triangle if and only if... ] a value of true is returned three )! 11, then I will purchase a computer of its components true but the is. Both are false shows you the notes and you do not see them, a value of q a. A declarative sentence which has one and only if q is false only when p and q are logically to... B and b are true or false a prime number ” and also “ q that! = b and b are true or false values years, 4 months ago two-way arrow )! \Sim p ) \Rightarrow r\ ) Solution and Password Submit let, a of! We ’ ll be going over how a table setup can help you remember the truth tables for statements! 9 years, 4 examples, and q is called the conclusion ( or 'if ' ) statements their... Statement ( pq ) ( qp ) is a quadrilateral is a conclusion p ↔ q is shown.! 'S call them p and q is always true also, when one is false, truth... It without using a Truth-Table out is the first row naturally follows this definition b b. About Us | Contact biconditional statement truth table | Facebook | Recommend this Page confident about your work. one can via. | Advertise with Us | Advertise with Us | Advertise with Us | Facebook | Recommend Page... Information and to our privacy policy they are of equal length a tautology is a and... With their truth-tables at BYJU 's definition of a biconditional statement is defined to true.: know how to do it without | {
"domain": "scenv.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9648551546097941,
"lm_q1q2_score": 0.8661459199556888,
"lm_q2_score": 0.8976952818435994,
"openwebmath_perplexity": 716.6132722582131,
"openwebmath_score": 0.46349072456359863,
"tags": null,
"url": "https://scenv.com/dreamcatcher-you-cqq/479aff-biconditional-statement-truth-table"
} |
at BYJU 's definition of a biconditional statement is defined to true.: know how to do it without using a Truth-Table x + 7 = 11 biconditional statement truth table. Password Submit for these statements have the same definition 3 ; Class -. Are listed in the next section or 'if ' ) statements and their from... A counter-example the front is true thus R is true no matter what they are of equal.! If q ' determining truth values following examples, and problem packs statements occur frequently in mathematics p and... Statements or events p and q is false only when p is false or the! To omit such columns if you are in Houston quadrilateral, then q immediately. Each and why it comes out the way it does we use the properties of logical equivalence to show this! A biconditional statement truth table from above are represented by a double-headed arrow ↔ ''. These two equivalent statements side by side in the first step of any table... P and q Example 1 helpful to think of the statement pq is false when... Rows to cover all possible scenarios will purchase a computer this section will... P → q is going to be true whenever both parts have the same truth value of q pq (. Other two types If-Then and if and only if q is true but the is. To your answer is provided in the first step of any truth table to determine how truth... That must be correct you make a truth table for p and q is false only when the is... Is there XNOR ( logical biconditional ) operator in c # triangle is isosceles if and only if both parts... Definition: a biconditional statement is logically equivalent to \ ( m \wedge p... Worksheets that get students ready for truth tables to determine the truth functional connective is. Are true form if and only if '', sometimes written as iff '' instead ...: if the polygon has only four sides, then the biconditional operator is by! Better understanding, you automatically know the other two types If-Then and if and only if +..., and problem packs raining and b are false | {
"domain": "scenv.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9648551546097941,
"lm_q1q2_score": 0.8661459199556888,
"lm_q2_score": 0.8976952818435994,
"openwebmath_perplexity": 716.6132722582131,
"openwebmath_score": 0.46349072456359863,
"tags": null,
"url": "https://scenv.com/dreamcatcher-you-cqq/479aff-biconditional-statement-truth-table"
} |
know the other two types If-Then and if and only if +..., and problem packs raining and b are false think of the:! Your answer is provided in the same truth value written as iff words, logical p... = 7 if and only if you make a truth table for p → q is false concept of truth! | {
"domain": "scenv.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9648551546097941,
"lm_q1q2_score": 0.8661459199556888,
"lm_q2_score": 0.8976952818435994,
"openwebmath_perplexity": 716.6132722582131,
"openwebmath_score": 0.46349072456359863,
"tags": null,
"url": "https://scenv.com/dreamcatcher-you-cqq/479aff-biconditional-statement-truth-table"
} |
Base Coat Paint Automotive, Outdoor Education Grants, Study Medicine In Russia On Scholarship, Hammered Spray Paint, Calcium Sulfate Dihydrate Uses, Cera Floor Mounted Wc, Embolism Vs Embolus, Chapter 4 Photosynthesis And Cellular Respiration Worksheets Answer Key, | {
"domain": "scenv.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9648551546097941,
"lm_q1q2_score": 0.8661459199556888,
"lm_q2_score": 0.8976952818435994,
"openwebmath_perplexity": 716.6132722582131,
"openwebmath_score": 0.46349072456359863,
"tags": null,
"url": "https://scenv.com/dreamcatcher-you-cqq/479aff-biconditional-statement-truth-table"
} |
# Definite integral of even powers of Cosine.
I'm looking for a step-by-step solution to the following integral, in terms of n$$\int_0^{\frac{\pi}{2}} \cos^{2n}(x) \ {dx}$$I actually KNOW that the solution is$${\frac{\pi}{2}} \prod_{k=1}^n \frac{2k-1}{2k}$$ But I would like to know how to get there. This is not homework, this is to further my own understanding. My own efforts to solve this consisted of expanding $\cos(x)$ in terms of $e^{ix}$ but this proved to be fruitless, leading me only to the following integral, where $u=e^{ix}$ $$-i\int_1^i \left({\frac{1+u^2}{2u}}\right)^{2n} \ {du}$$
• Use the binomial expansion of the integrand. You can easily obtain an answer in terms of a finite summation. – Mark Viola Jun 23 '15 at 22:33
• And in that summation all but one term has integral $0$. At least after the right initial step... – David C. Ullrich Jun 23 '15 at 22:35
• @Dr.MV, you are referring to the second integral in terms of u, correct? – JacksonFitzsimmons Jun 23 '15 at 22:36
• You'll have an answer as fast as I can type it. Hang on... – David C. Ullrich Jun 23 '15 at 22:38
• – Lucian Jun 24 '15 at 1:21
There are several approaches. One, that has been given in previous answers, uses the fact that $\cos(\theta)=\frac12(e^{i\theta}+e^{i\theta})$ and that $\int_0^{2\pi}e^{in\theta}\,\mathrm{d}\theta=2\pi$ if $n=0$, and vanishes otherwise, to get \begin{align} \int_0^{\pi/2}\cos^{2n}(\theta)\,\mathrm{d}\theta &=\frac14\int_0^{2\pi}\cos^{2n}(\theta)\,\mathrm{d}\theta\\ &=\frac1{4^{n+1}}\int_0^{2\pi}\left(e^{i\theta}+e^{i\theta}\right)^{2n}\,\mathrm{d}\theta\\ &=\frac{2\pi}{4^{n+1}}\binom{2n}{n} \end{align} | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429624315188,
"lm_q1q2_score": 0.8661256064213304,
"lm_q2_score": 0.8807970764133561,
"openwebmath_perplexity": 708.964052480164,
"openwebmath_score": 0.9242236018180847,
"tags": null,
"url": "https://math.stackexchange.com/questions/1336903/definite-integral-of-even-powers-of-cosine"
} |
Another approach is to integrate by parts \begin{align} \int_0^{\pi/2}\cos^{2n}(\theta)\,\mathrm{d}\theta &=\int_0^{\pi/2}\cos^{2n-1}(\theta)\,\mathrm{d}\sin(\theta)\\ &=(2n-1)\int_0^{\pi/2}\sin^2(\theta)\cos^{2n-2}(\theta)\,\mathrm{d}\theta\\ &=(2n-1)\int_0^{\pi/2}\left(\cos^{2n-2}(\theta)-\cos^{2n}(\theta)\right)\,\mathrm{d}\theta\\ &=\frac{2n-1}{2n}\int_0^{\pi/2}\cos^{2n-2}(\theta)\,\mathrm{d}\theta\\ \end{align} and use induction to get $$\int_0^{\pi/2}\cos^{2n}(\theta)\,\mathrm{d}\theta =\frac\pi2\prod_{k=1}^n\frac{2k-1}{2k}$$
Note that \begin{align} \frac\pi2\prod_{k=1}^n\frac{2k-1}{2k} &=\frac\pi2\frac{(2n)!}{(2^nn!)^2}\\ &=\frac{\pi}{2^{2n+1}}\binom{2n}{n} \end{align}
• Thank you for giving me multiple methods. – JacksonFitzsimmons Jun 23 '15 at 23:23
• You're welcome. I started writing my answer before Bernard posted his answer, which also uses integration by parts. – robjohn Jun 23 '15 at 23:30
Integrate by parts, setting $$u=\cos^{2n-1}x,\enspace \operatorname{d}\mkern-2mu v=\cos x\operatorname{d}\mkern-2mu x,\enspace\text{whence}\quad \operatorname{d}\mkern-2mu u=-(2n-1)\cos^{2n-2}x \operatorname{d}\mkern-2mux,\enspace v= \sin x$$ Let's call $I_{2n}$ the integral. One obtains: $$I_{2n}=\Bigl[\sin x\mkern1mu\cos^{2n-1}x\Bigr]_0^{\tfrac\pi2}+(2n-1)(I_{2n-2}-I_{2n})=(2n-1)(I_{2n-2}-I_{2n})$$ whence the recurrence relation: $$I_{2n}=\frac{2n-1}{2n}I_{2n-2}.$$ Now write all these relations down to $n=1$, multiply the equalities thus obtained and simplify.
First, note that $\int_0^{\pi/2}=\frac14\int_0^{2\pi}$ by various symmetries.
Now say $\cos(t)=\frac12(e^{it}+e^{-it})$ and apply the binomial theorem. You get terms consisting of various powers of $e^{it}$. All those terms have integral $0$ except the middle one: $(e^{it})^n(e^{-it})^n=1$. So you get $$\frac14\int_0^{2\pi}\cos^{2n}(t)\,dt=\frac142^{-2n}(2\pi)C(2n,n),$$where $C()$ is a binomial coefficient. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429624315188,
"lm_q1q2_score": 0.8661256064213304,
"lm_q2_score": 0.8807970764133561,
"openwebmath_perplexity": 708.964052480164,
"openwebmath_score": 0.9242236018180847,
"tags": null,
"url": "https://math.stackexchange.com/questions/1336903/definite-integral-of-even-powers-of-cosine"
} |
The answer you want must now follow by induction (unless I dropped a factor, in which case it follows by induction from the corrected version of the above).
• It's nice to see you again. It's been a while. – robjohn Jun 23 '15 at 23:30
• Nice to be seen... – David C. Ullrich Jun 23 '15 at 23:34
First, note that it is:
$$\frac{1}{4}\int_{0}^{2\pi} \cos^{2n}x\,dx$$
Now $$\cos^{2n}(x)=\frac{1}{2^{2n}}\left(e^{ix}+e^{-ix}\right)^{2n}$$. And for integer $m\neq 0$, $\int_{0}^{2\pi}e^{imx}\,dx = 0$.
So you only care about the constant term of $(e^{ix}+e^{-ix})^{2n}$, which is $\binom{2n}{n}$.
So the integral is:
$$\frac{1}{4}\cdot 2\pi \cdot \frac{1}{2^{2n}} \binom{2n}{n}=\frac{\pi}{2}\frac{1}{2^{2n}}\binom{2n}{n}$$
Then prove that $$\frac{\binom{2n}{n}}{2^{2n}} = \prod_{k=1}^n \frac{2k-1}{2k}$$
You can prove this last by induction:
$$\binom{2(n+1)}{n+1} = \frac{(2n+1)(2n+2)}{(n+1)(n+1)}\binom{2n}{n}=2\frac{2(n+1)-1}{2(n+1)}\binom{2n}{n}$$
• Hum constant term of $\left(e^{ix}+e^{-ix}\right)^{2n}$ is $\binom{2n}{n}$, can you explain me ? ^^ – ParaH2 Jun 23 '15 at 22:48
• If you expand $(a+a^{-1})^{2n}$ via the binomial theorem, then the constant terms is $\binom{2n}{n}$. All the other terms are of the form $Ca^k$ for $k\neq 0$. Not sure how to make that more clear. – Thomas Andrews Jun 23 '15 at 22:49
• Ok, and why are you seeking constants terms ? – ParaH2 Jun 23 '15 at 22:51
• Because the integrals of all the other terms are zero, as I noted. So the only term that contributes a non-zero value to the integral is the constant term. @Shadock – Thomas Andrews Jun 23 '15 at 22:52
• OK thank you, I'm a chemist, I have some knowlegde in maths because i love them, I'm ridiculous here haha ^^ – ParaH2 Jun 23 '15 at 22:55
You were on the right track using Euler's Identity and writing $\cos x=\frac12(e^{ix}+e^{-ix})$. Proceeding accordingly we have, | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429624315188,
"lm_q1q2_score": 0.8661256064213304,
"lm_q2_score": 0.8807970764133561,
"openwebmath_perplexity": 708.964052480164,
"openwebmath_score": 0.9242236018180847,
"tags": null,
"url": "https://math.stackexchange.com/questions/1336903/definite-integral-of-even-powers-of-cosine"
} |
\begin{align} \int_0^{\pi/2}\cos^{2n}x\,dx&=\int_0^{\pi/2}\left(\frac{e^{ix}+e^{-ix}}{2}\right)^{2n}dx\\\\ &=\frac{1}{4^n}\int_0^{\pi/2}\sum_{k=0}^{2n}\binom{2n}{k}e^{ikx}e^{-ix(2n-k)}dx\\\\ &=\frac{1}{4^n}\sum_{k=0}^{2n}\binom{2n}{k}\int_0^{\pi/2}e^{ix2(k-n)}dx\\\\ &=\frac{1}{4^n}\sum_{k=0}^{2n}\binom{2n}{k}\frac{\pi}{2}\delta_{nk}+\frac{1}{4^n}\sum_{k=0, k\ne n}^{2n}\binom{2n}{k}\frac{(-1)^{n-k}-1}{i2(k-n)} \tag 1\\\\ &=\frac{\pi}{2}\frac{1}{4^n}\binom{2n}{n}\\\\ &=\frac{\pi}{2}\frac{1}{4^n}\frac{(2n)!}{(n!)^2}\\\\ &=\frac{\pi}{2}\left(\frac{1}{2^n\,n!}\right)\left(\frac{(2n)!}{2^n\,n!}\right)\\\\ &=\frac{\pi}{2}\left(\frac{1}{(2n)!!}\right)\left((2n-1)!!\right)\\\\ &=\frac{\pi}{2}\frac{(2n-1)!!}{(2n)!!}\\\\ &=\frac{\pi}{2}\prod_{k=1}^{n}\frac{2k-1}{2k} \end{align}
as was to be shown!!
Note that the second sum in $(1)$ is purely imaginary and, thereby, must vanish. One can easily show it vanishes by exploiting symmetry. We now explicitly show this.
\begin{align} \sum_{k=0, k\ne n}^{2n}\binom{2n}{k}\frac{(-1)^{n-k}-1}{i2(k-n)} &=\sum_{k=0}^{n-1}\binom{2n}{k}\frac{(-1)^{n-k}-1}{i2(k-n)}+\sum_{k=n+1}^{n-1}\binom{2n}{k}\frac{(-1)^{n-k}-1}{i2(k-n)}\\\\ &=\sum_{k=0}^{n-1}\binom{2n}{k}\frac{(-1)^{n-k}-1}{i2(k-n)}+\sum_{m=0}^{n-1}\binom{2n}{2n-m}\frac{(-1)^{m-n}}{i2(n-m)} \text{substituting m=2n-k} \\\\ &=\sum_{k=0}^{n-1}\binom{2n}{k}\frac{(-1)^{n-k}-1}{i2(k-n)}-\sum_{k=0}^{n-1}\binom{2n}{2n-k}\frac{(-1)^{k-n}}{i2(k-n)} \\\\ &=\sum_{k=0}^{n-1}\binom{2n}{k}\frac{(-1)^{n-k}-1}{i2(k-n)}-\sum_{k=0}^{n-1}\binom{2n}{k}\frac{(-1)^{n-k}}{i2(k-n)} \text{Using}\,\, \binom{2n}{2n-k}= \binom{2n}{k} \\\\ &=0 \end{align}
as was to be shown! | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429624315188,
"lm_q1q2_score": 0.8661256064213304,
"lm_q2_score": 0.8807970764133561,
"openwebmath_perplexity": 708.964052480164,
"openwebmath_score": 0.9242236018180847,
"tags": null,
"url": "https://math.stackexchange.com/questions/1336903/definite-integral-of-even-powers-of-cosine"
} |
as was to be shown!
• $\delta_{nk}$ ? – ParaH2 Jun 23 '15 at 22:43
• Why is the integral from $$\int_{0}^{\pi/2}e^{2(k-n)ix}\,dx=\frac{\pi}{2}\delta_{nk}?$$ It's true for $\int_{0}^{\pi}$ or $\int_0^{2\pi}$, but not sure if for $\int_0^{\pi/2}$. – Thomas Andrews Jun 23 '15 at 22:44
• The Kronecker Delta. It is $1$ when the indices are equal and $0$ otherwise. – Mark Viola Jun 23 '15 at 22:44
• $\delta_{nk}=\delta_{n,k}=1$ if $n=k$, $0$ otherwise. – David C. Ullrich Jun 23 '15 at 22:45
• @ThomasAndrews All of the other terms are purely imaginary and cancel by symmetry. – Mark Viola Jun 23 '15 at 22:45 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429624315188,
"lm_q1q2_score": 0.8661256064213304,
"lm_q2_score": 0.8807970764133561,
"openwebmath_perplexity": 708.964052480164,
"openwebmath_score": 0.9242236018180847,
"tags": null,
"url": "https://math.stackexchange.com/questions/1336903/definite-integral-of-even-powers-of-cosine"
} |
# Rolle's theorem: what's the right statement of the theorem?
In the fourth edition of "Introduction to Real Analysis" by Bartle and Sherbert, theorem 6.2.3 (Rolle's theorem) states,
Suppose that f is continuous on a closed interval $I := [a, b]$, that the derivative of $f$ exists at every point of the open interval $(a, b)$, and that $f(a) = f(b) = 0$. Then there exists at least one point $c$ in $(a, b)$ such that the derivative of $f$ is zero at $c$.
Now, why are we taking $f(a)=0=f(b)$? Is $f(a)=f(b)$ not sufficient?
You are right, taking $f(a) = f(b)$ is sufficient.
But, one can prove the theorem in this general scenario using the theorem for the case $f(a) = 0 = f(b)$, as follows:
Assume Rolle's theorem as stated in the question details is true. Let $f$ be a function satisfying the same hypotheses, except that $f(a) = f(b) = k$, where $k$ is not necessarily equal to zero. Then, the function $g(x) = f(x) - k$ satisfies the hypotheses of Rolle's theorem, and so there is a point $c$ such that $g'(c) = 0$. But $g'(c) = f'(c)$, so we are done.
So, it doesn't really matter which one we use, as both versions are seen to be equivalent to each other.
• If by 'right statement' OP means 'most general', then no we don't need f(a),f(b) to equal zero. Better to state in the more general. And Rolle's Theorem not always about roots (f(a)=f(b)=0). The version I was taught was simply a formalization that between two points with equal y-value, there is a peak with zero derivative. – smci Aug 5 '18 at 23:51
• This could be an answer on its own. – user279515 Aug 6 '18 at 3:55
Yes, you are right: $f(a)=f(b)$ is enough. However, it is usual to add the condition $=0$ after $f(a)=f(b)$ so that the theorem becomes: between two roots of $f$, there's a root of $f'$, which is the original theorem due Michel Rolle (who stated it for polynomials only).
Anyway, this is a moot point, since both statements are just a particular case of the Mean Value Theorem. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.972830769252026,
"lm_q1q2_score": 0.866105620398582,
"lm_q2_score": 0.8902942297605357,
"openwebmath_perplexity": 191.14458380977035,
"openwebmath_score": 0.9453277587890625,
"tags": null,
"url": "https://math.stackexchange.com/questions/2872218/rolles-theorem-whats-the-right-statement-of-the-theorem"
} |
• OTOH, the usual way to prove the Mean Value Theorem is by applying the theorem of Rolle. Depending on how much one 'groks' working with derivatives, the step from the general ($f(a) = f(b)$ but not necessarily $=0$) version of Rolle to the Mean Value Theorem is not much bigger than from the "$f(a) = f(b)=0$" version of Rolle to the general one. – Ingix Aug 4 '18 at 17:32
The two versions are clearly equivalent. Suppose just $f(a)=f(b)$. Let $g(t)=f(t)-f(a)$. Then $g(a)=g(b)=0$, so the formally weaker version shows $g'(c)=0$, hence $f'(c)=0$ for some $c$. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.972830769252026,
"lm_q1q2_score": 0.866105620398582,
"lm_q2_score": 0.8902942297605357,
"openwebmath_perplexity": 191.14458380977035,
"openwebmath_score": 0.9453277587890625,
"tags": null,
"url": "https://math.stackexchange.com/questions/2872218/rolles-theorem-whats-the-right-statement-of-the-theorem"
} |
# Probability that the two segments intersect
P and Q are uniformly distributed in a square of side AB. What is the probability that segments AP and BQ intersect?
Inspired by Lee David Chung Lin's answer.
Consider the four events $$AP \cap BQ \neq \varnothing, \\ AP \cap DQ \neq \varnothing, \\ CP \cap BQ \neq \varnothing, \\ CP \cap DQ \neq \varnothing.$$
Ignoring degenerate configurations, the events are mutually disjoint, equiprobable, and cover the whole space. $p = 1/4$.
• This is really great. The reason I felt it was necessary to awkwardly make rigorous the $1/4$ (argument in the last part of my post), was exactly due to the lack of symmetry on the $AB$-and-$PQ$ configuration with respect to the square. I think this is THE canonical treatment of it as a geometric probability problem. May 2, 2018 at 8:14
Suppose we start with $Q = (x, y)$. The admissible positions for $P$ will be in the triangle $BQR$ if $y < x$ or in the quadrilateral $BQRC$ if $y > x$. The areas are calculated from the sides and the heights, and $$p = \int_0^1 \int_0^x \frac {y (1 - x)} {2 x} dy dx + \int_0^1 \int_x^1 \left( 1 - \frac x {2 y} - \frac y 2 \right) dy dx = \frac 1 4.$$
Here's a calculation free "cut-and-paste" argument. Feel free to skip the words and directly examine the figures.
Consider the point $Q$ only in a quarter of the square $\square ABCD$, as shown in the left plot below. We will combine the "region of $P$ that creates a crossing" associated with $Q$ and its 4-fold symmetric mirror images.
As the right plot below shows, point $Q'$ is reflected with respect to (WRT) the main diagonal $\overline{AC}$, point $q$ is reflected WRT the off-diagonal $\overline{BD}$, and point $q'$ is doubly-reflected (equivalent to rotated $180^{\circ}$). The region that is admissible (borrowing the great term from @Maxim) associated with $Q$ is highlighted below on the left (in magenta), and the region for $Q'$ is highlighted on the right plot (in blue). | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9908743642277194,
"lm_q1q2_score": 0.8661007360559426,
"lm_q2_score": 0.8740772466456689,
"openwebmath_perplexity": 390.5814220446503,
"openwebmath_score": 0.8973247408866882,
"tags": null,
"url": "https://math.stackexchange.com/questions/2759011/probability-that-the-two-segments-intersect/2759538"
} |
The labels for corner points $A,B,C,D$ will be omitted in the plots from now on, because they are unnecessary most of the time and a bit distracting. We can cut-and-paste the triangular blue "$Q'$ region" in the manner demonstrated below, thanks to the mirror symmetry by construct of $Q'$.
Note that the resultant quadrilateral consists of an obtuse triangle ($\triangle DQB$) plus THE isosceles right triangle, which is half of $\square ABCD$. Similarly, the admissible regions associated with $q$ and $q'$ can be cut-and-paste into a quadrilateral that is half of $\square ABCD$ minus an obtuse triangle ($\triangle DqB$).
I sincerely apologize for the poor choice of colors if the readers find the figures unclear.
Due to the symmetry WRT the off-diagonal, we can cut-and-paste (flip) the combined-$(q,q')$-region and have the exact full square.
In short, since we effectively quadrupled the admissible region to arrive at $1$ (unity), the actual probability is $\displaystyle \frac14$.
If one is unsatisfied with the brief ending remark just above, below is the more detailed explanation.
Formally, instead of scanning point $Q$ over the entire square, we scan it only in a quadrant and reallocate the probability mass from the 3 mirrored positions $(Q',q,q')$. That is, we redefine the mass associated with $(Q',q,q')$ as the contribution associated with $Q$.
By reallocation, it means that the (conditional) probability of crossing is to be defined as zero when $Q$ is outside of the quadrant.
After the reallocation, when $Q$ is in the quadrant, the (conditional) probability of making an intersection is $1$, because $\square ABCD$ (result of cut-and-paste) divided by $\square ABCD$ (the domain of point $P$) is one.
In other words, the re-distributed conditional probability is $1$ over one-fourth of the domain (of $Q$) and $0$ elsewhere. Thus the overall probability is one-fourth. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9908743642277194,
"lm_q1q2_score": 0.8661007360559426,
"lm_q2_score": 0.8740772466456689,
"openwebmath_perplexity": 390.5814220446503,
"openwebmath_score": 0.8973247408866882,
"tags": null,
"url": "https://math.stackexchange.com/questions/2759011/probability-that-the-two-segments-intersect/2759538"
} |
• A really nice answer. I suppose the last part can also be explained by saying that when computing the integral sums $S(x, y) \Delta x \Delta y$ over a symmetric grid, we can combine the four related terms, and the result will be the same as summing $\Delta x \Delta y / 4$. I've added another answer, exploiting the symmetry over the vertices of the square instead of the symmetries over the positions of $P$ and $Q$. May 2, 2018 at 8:03 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9908743642277194,
"lm_q1q2_score": 0.8661007360559426,
"lm_q2_score": 0.8740772466456689,
"openwebmath_perplexity": 390.5814220446503,
"openwebmath_score": 0.8973247408866882,
"tags": null,
"url": "https://math.stackexchange.com/questions/2759011/probability-that-the-two-segments-intersect/2759538"
} |
# Probability for #suits in 5 card poker hand
Given a 5 card poker hand from a standard deck, I'm looking to calculate the probability of getting: all 1 suit, 2 different suits, 3 different suits or 4 different suits. All one suit is straight-forward - $\frac{\binom{13}{5}*\binom{4}{1}}{\binom{52}{5}}$- pick five different ranks, each from the same suit.
Likewise, 4 seems fairly simple: $\frac{\binom{4}{1}\binom{13}{2}\binom{13}{1}^3}{\binom{52}{5}}$ - pick one suit to grab two cards from, then pick one card from each other suit.
Its on 2 and 3 that I get kind of stuck - I'm not sure how to set them up! I don't see why something along the lines of
$\frac{\binom{4}{2}*\binom{26}{6} - \binom{13}{5}*\binom{4}{1}}{\binom{52}{5}}$ doesn't work for 2 suits; i.e. picking 2 suits, choose 5 cards, subtracting off the ways in which you could end up with one suit. Similarly, for 3 I would expect
$\frac{\binom{4}{3}*\binom{39}{5}-\binom{4}{2}*\binom{26}{5}}{\binom{52}{5}}$ to give the answer (picking 5 cards from the group containing 3 suits, subtracting off those hands with fewer than 3 suits), but if I sum the probabilities it comes out incorrectly.
Thank you very much for your help! | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9908743634192694,
"lm_q1q2_score": 0.8661007353492949,
"lm_q2_score": 0.8740772466456689,
"openwebmath_perplexity": 370.4676223005469,
"openwebmath_score": 0.7736230492591858,
"tags": null,
"url": "http://math.stackexchange.com/questions/224974/probability-for-suits-in-5-card-poker-hand"
} |
Thank you very much for your help!
-
Perhaps there is an easier way, but could always count the number of ways of getting 1 club and 4 diamonds, 2 clubs and 3 diamonds, 3 clubs and 2 diamonds, 4 clubs and 1 diamond. Then, multiply your answer by ${4 \choose 2}$. – JavaMan Oct 30 '12 at 5:22
While this technique makes sense to me, I fear it would translate poorly to the 3 suit question - i.e. 3 clubs, one heart, one spade; 2 clubs, 2 hearts, 1 spade; 1 club, 2 hearts, 2 spades etc... seems that it could get tedious/confusing quickly. I feel like there must be a simpler way to go about it! – jeliot Oct 30 '12 at 5:31
Actually, the three suit case isn't so bad: Let $(a,b,c)$ denote the permissible number of clubs, diamonds, and hearts. Then you have to consider $(3,1,1); (1,3,1); (3,1,1); (1,2,2); (2,1,2); (2,2,1)$. – JavaMan Oct 30 '12 at 6:09
Whoops, that's really not too intimidating at all! mjqxxxx offers an excellent explanation for a slightly more concise formula down below as well :) – jeliot Oct 30 '12 at 7:35 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9908743634192694,
"lm_q1q2_score": 0.8661007353492949,
"lm_q2_score": 0.8740772466456689,
"openwebmath_perplexity": 370.4676223005469,
"openwebmath_score": 0.7736230492591858,
"tags": null,
"url": "http://math.stackexchange.com/questions/224974/probability-for-suits-in-5-card-poker-hand"
} |
The "exclusion-based" formulation you're trying to use for exactly two suits works fine; you just have a minor error. You can choose the two suits in $4\choose 2$ ways, and then choose a five-card hand from the $26$ cards in those two suits in $26\choose 5$ ways. This counts a number of single-suit hands as well, which you want to exclude. In fact, each of the $4\times{13\choose 5}$ single-suit hands is included exactly $3$ times in your original count (once with each possible "partner suit"). After excluding these with the correct multiplicity, you have $${4\choose 2}{26\choose 5}-3{4\choose 1}{13\choose 5}=379236$$ hands containing exactly two suits. As a double-check, you can calculate from the other direction ("inclusion-based"). A hand with two suits has either four cards from one suit and one from the other, or three cards from one suit and two from the other. There are $12$ ways to choose the major and minor suits. For each (ordered) pair of suits, there are ${13\choose 4}{13\choose 1} + {13\choose 3}{13 \choose 2}=31603$ ways to make a hand; the result is $12\times 31603 = 379236$ again. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9908743634192694,
"lm_q1q2_score": 0.8661007353492949,
"lm_q2_score": 0.8740772466456689,
"openwebmath_perplexity": 370.4676223005469,
"openwebmath_score": 0.7736230492591858,
"tags": null,
"url": "http://math.stackexchange.com/questions/224974/probability-for-suits-in-5-card-poker-hand"
} |
With the two-suit result in hand, you can finish the problem. The number of single-suit hands, as you argued, is $4\times{13\choose 5}=5148.$ The number of four-suit hands is $4\times 13^3\times{13\choose 2}=685464.$ Therefore, the number of three-suit hands (the only remaining possibility) is ${52\choose 5}-5148-379236-685464=1529112.$ To double-check this, note that a three-suit hand has $(1,2,2,0)$ or $(3,1,1,0)$ cards per suit. There are $12$ ways to choose the "loner" (first) and "excluded" (last) suits. For each (ordered) pair of suits, there are $13\times{13\choose 2}^2 + {13\choose 3}\times 13^2=127426$ ways to make a hand; the result is $12\times 127426 = 1529112$ again. The associated probabilities are: $$\begin{eqnarray} P_1 &=& \frac{5148}{2598960} &=& 0.198\% \\ P_2 &=& \frac{379236}{2598960} &=& 14.592\% \\ P_3 &=& \frac{1529112}{2598960} &=& 58.836\% \\ P_4 &=& \frac{685464}{2598960} &=& 26.375\% \\ \end{eqnarray}$$
-
Thank you so much - perfectly explained! – jeliot Oct 31 '12 at 0:24
As an example of another technique, which is useful for counting poker hands, particularly two pairs, we count the $3$-suit hands.
There are two types of $3$-suit hand: $3$-$1$-$1$ and $2$-$2$-$1$.
We count the $3$-$1$-$1$ hands. The suit we have $3$ of can be chosen in $\dbinom{4}{1}$ ways. For each such way, the actual cards can be chosen in $\dbinom{13}{3}$ ways. Now the suits we have $1$ of can be chosen in $\dbinom{3}{2}$ ways, and the actual cards in $\dbinom{13}{1}^2$ ways, for a total of $$\binom{4}{1}\binom{13}{3}\binom{3}{2}\binom{13}{1}^2.$$
Next we count the $2$-$2$-$1$ hands. The suit in which we have a singleton can be chosen in $\dbinom{4}{1}$ ways. For each such choice, the actual card can be chosen in $\dbinom{13}{1}$ ways. Now the suits we have doubletons in can be chosen in $\dbinom{3}{2}$ ways, and the actual cards in $\dbinom{13}{2}^2$ ways, for a total of $$\binom{4}{1}\binom{13}{1}\binom{3}{2}\binom{13}{2}^2.$$
- | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9908743634192694,
"lm_q1q2_score": 0.8661007353492949,
"lm_q2_score": 0.8740772466456689,
"openwebmath_perplexity": 370.4676223005469,
"openwebmath_score": 0.7736230492591858,
"tags": null,
"url": "http://math.stackexchange.com/questions/224974/probability-for-suits-in-5-card-poker-hand"
} |
GMAT Question of the Day - Daily to your Mailbox; hard ones only
It is currently 16 Oct 2019, 10:08
### GMAT Club Daily Prep
#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You
we will pick new questions that match your level based on your Timer History
Track
every week, we’ll send you an estimated GMAT score based on your performance
Practice
Pays
we will pick new questions that match your level based on your Timer History
# A discount electronics store normally sells all merchandise
Author Message
TAGS:
### Hide Tags
Intern
Joined: 05 Nov 2011
Posts: 15
Location: France, Metropolitan
A discount electronics store normally sells all merchandise [#permalink]
### Show Tags
20 Feb 2015, 02:32
2
2
00:00
Difficulty:
25% (medium)
Question Stats:
78% (01:56) correct 22% (01:54) wrong based on 103 sessions
### HideShow timer Statistics
A discount electronics store normally sells all merchandise at a discount of 10 percent to 30 percent off the suggested retail price. If, during a special sale, an additional 20 percent were to be deducted from the discount price, what would be the lowest possible price of an item costing $260 before any discount? (A)$130.00
(B) $145.60 (C)$163.80
(D) $182.00 (E)$210.00
Intern
Joined: 05 Nov 2011
Posts: 15
Location: France, Metropolitan
A discount electronics store normally sells all merchandise [#permalink]
### Show Tags
20 Feb 2015, 02:45
1
Stardust Chris wrote:
A discount electronics store normally sells all merchandise at a discount of 10 percent to 30 percent off the suggested retail price. If, during a special sale, an additional 20 percent were to be deducted from the discount price, what would be the lowest possible price of an item costing $260 before any discount? (A)$130.00
(B) $145.60 (C)$163.80
(D) $182.00 (E)$210.00 | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9678992969868542,
"lm_q1q2_score": 0.8660661013019167,
"lm_q2_score": 0.8947894724152068,
"openwebmath_perplexity": 11828.661582317838,
"openwebmath_score": 0.32809552550315857,
"tags": null,
"url": "https://gmatclub.com/forum/a-discount-electronics-store-normally-sells-all-merchandise-193516.html"
} |
(B) $145.60 (C)$163.80
(D) $182.00 (E)$210.00
Original price : 260 $Max first discount = -30% Thus : $$260*(1-\frac{30}{100})=182$$ Second discount on the discounted price = -20% Thus : $$182*(1-\frac{20}{100})=145,6$$ Answer B. EMPOWERgmat Instructor Status: GMAT Assassin/Co-Founder Affiliations: EMPOWERgmat Joined: 19 Dec 2014 Posts: 15262 Location: United States (CA) GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: A discount electronics store normally sells all merchandise [#permalink] ### Show Tags 20 Feb 2015, 13:59 2 Hi Stardust Chris, Since the question is essentially just about multiplication, you can do the various math "steps" in a variety of ways (depending on whichever method you find easiest). We're told that the first discount is 10% to 30%, inclusive. We're told that the next discount is 20% off of the DISCOUNTED price.... We're told to MAXIMIZE the discount (thus, 30% off the original price and then 20% off of the discounted price). That "math" can be written in a number of different ways (fractions, decimals, etc.): 30% off = (1 - .3) = (1 - 30/100) = (.7) and the same can be done with the 20% additional discount... The final price of an item that originally cost$260 would be.....
($260)(.7)(.8) = ($260)(.56) | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9678992969868542,
"lm_q1q2_score": 0.8660661013019167,
"lm_q2_score": 0.8947894724152068,
"openwebmath_perplexity": 11828.661582317838,
"openwebmath_score": 0.32809552550315857,
"tags": null,
"url": "https://gmatclub.com/forum/a-discount-electronics-store-normally-sells-all-merchandise-193516.html"
} |
($260)(.7)(.8) = ($260)(.56)
Since .56 is a little more than 1/2, we're looking for an answer that's a little more than $130.... Since these answer choices are so "spread out", we don't really have to do the calculation... Final Answer: GMAT assassins aren't born, they're made, Rich _________________ Contact Rich at: Rich.C@empowergmat.com The Course Used By GMAT Club Moderators To Earn 750+ souvik101990 Score: 760 Q50 V42 ★★★★★ ENGRTOMBA2018 Score: 750 Q49 V44 ★★★★★ Board of Directors Status: QA & VA Forum Moderator Joined: 11 Jun 2011 Posts: 4782 Location: India GPA: 3.5 WE: Business Development (Commercial Banking) Re: A discount electronics store normally sells all merchandise [#permalink] ### Show Tags 16 Jan 2018, 11:01 Stardust Chris wrote: A discount electronics store normally sells all merchandise at a discount of 10 percent to 30 percent off the suggested retail price. If, during a special sale, an additional 20 percent were to be deducted from the discount price, what would be the lowest possible price of an item costing$260 before any discount?
(A) $130.00 (B)$145.60
(C) $163.80 (D)$182.00 | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9678992969868542,
"lm_q1q2_score": 0.8660661013019167,
"lm_q2_score": 0.8947894724152068,
"openwebmath_perplexity": 11828.661582317838,
"openwebmath_score": 0.32809552550315857,
"tags": null,
"url": "https://gmatclub.com/forum/a-discount-electronics-store-normally-sells-all-merchandise-193516.html"
} |
(A) $130.00 (B)$145.60
(C) $163.80 (D)$182.00
(E) $210.00 Max possible value after first discount is 70% of 260 = 182 Max possible value after second discount is 80% of 182 = 145.60 Thus, answer will be (B) 145.60 _________________ Thanks and Regards Abhishek.... PLEASE FOLLOW THE RULES FOR POSTING IN QA AND VA FORUM AND USE SEARCH FUNCTION BEFORE POSTING NEW QUESTIONS How to use Search Function in GMAT Club | Rules for Posting in QA forum | Writing Mathematical Formulas |Rules for Posting in VA forum | Request Expert's Reply ( VA Forum Only ) Target Test Prep Representative Status: Founder & CEO Affiliations: Target Test Prep Joined: 14 Oct 2015 Posts: 8069 Location: United States (CA) Re: A discount electronics store normally sells all merchandise [#permalink] ### Show Tags 18 Jan 2018, 14:16 Stardust Chris wrote: A discount electronics store normally sells all merchandise at a discount of 10 percent to 30 percent off the suggested retail price. If, during a special sale, an additional 20 percent were to be deducted from the discount price, what would be the lowest possible price of an item costing$260 before any discount?
(A) $130.00 (B)$145.60
(C) $163.80 (D)$182.00
(E) \$210.00
The lowest possible price would be:
260 x 0.7 x 0.8
182 x 0.8 = 145.60
_________________
# Scott Woodbury-Stewart
Founder and CEO
Scott@TargetTestPrep.com
122 Reviews
5-star rated online GMAT quant
self study course
See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews
If you find one of my posts helpful, please take a moment to click on the "Kudos" button.
Manager
Joined: 03 Sep 2018
Posts: 174
Re: A discount electronics store normally sells all merchandise [#permalink]
### Show Tags
26 Aug 2019, 01:41
260*(30+20+(30*20)/100)%=145.6
Re: A discount electronics store normally sells all merchandise [#permalink] 26 Aug 2019, 01:41
Display posts from previous: Sort by | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9678992969868542,
"lm_q1q2_score": 0.8660661013019167,
"lm_q2_score": 0.8947894724152068,
"openwebmath_perplexity": 11828.661582317838,
"openwebmath_score": 0.32809552550315857,
"tags": null,
"url": "https://gmatclub.com/forum/a-discount-electronics-store-normally-sells-all-merchandise-193516.html"
} |
# Another puzzle with area
Squares $$ABCD, DCGH, BEFG$$ and $$ELKM$$ are positioned as shown on the picture. Find the area of triangle $$DGK$$ if you know that the area of square $$ABCD$$ is $$20$$.
Here's another solution using the fact that $$\triangle ABC$$ and $$\triangle ABD$$ have the same area if $$AB \parallel CD$$:
The reason is that $$CE = DF$$, so $$\frac12 AB \cdot CE = \frac12 AB \cdot DF$$.
Now, since $$DG \parallel EK$$ and $$BD \parallel EG$$, we have $$\triangle DGK = \triangle DEG = \triangle BEG$$ in area:
The result is hence
$$2 \cdot 20 = 40$$.
• Yes, well this is basicly the same solution as hexomino's. Apr 17 at 6:15
• @Greedoid It may use the same principle as Hex's proof, but it applies that principle in a different manner resulting in different triangles. Apr 17 at 7:20
Here is a quick way to do it
The locus of the point $$K$$ as the side length of the square $$EMKL$$ varies is a line which is parallel to $$DG$$. Hence, if we consider $$DG$$ as the base of the triangle $$DGK$$ then its perpendicular height, and thus its area, is invariant with respect to the size of the square $$EMKL$$.
This means that we could perform the calculation with the side length $$EM$$ being anything we like and the answer would be the same. If we make it so that the point $$M$$ coincides with the point $$F$$, it is easy to compute that the area of triangle $$DGK$$ is twice the area of the square $$ABCD$$ and thus the answer is $$40$$.
• I like this puzzle - at first sight there is a constraint missing but the answer comes out cleanly :) I think this puzzle would be suitable for some math channel such as 3Blue1Brown Apr 17 at 0:01
• Not worth a separate answer but I'd say the most opportunistic placement of M would be as the midpoint of E and F. Apr 17 at 7:37
Well, this answer is a bit late, but still quite simple I believe: | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9796676466573412,
"lm_q1q2_score": 0.8660646946729094,
"lm_q2_score": 0.8840392939666336,
"openwebmath_perplexity": 126.29980772497434,
"openwebmath_score": 0.9412528872489929,
"tags": null,
"url": "https://puzzling.stackexchange.com/questions/109512/another-puzzle-with-area"
} |
Well, this answer is a bit late, but still quite simple I believe:
Let's introduce a coordinate system with origin at $$A$$, having $$AL$$ and $$AH$$ as $$x$$ and $$y$$ axes respectively. Also, let $$AB=1$$ (well, the $$ABCD$$ square has now an area of $$1$$, not $$20$$, but we can just scale the final result 20 times, i.e. assuming that we scaled down the entire picture $$\sqrt{20}$$ times on each axis - otherwise we had to deal with a bunch of square roots). Now, $$D$$ has coordinates $$(0, 1)$$, $$G$$ is $$(1,2)$$ (since the squares $$ABCD$$ and $$DCGH$$ are equal with side $$1$$), and $$K$$ is $$(s + 3, s)$$ where $$s$$ is the side of $$ELKM$$ square, which we do not know. (The $$BEFG$$ square must have the side of $$2$$, because $$BG=BC+CG=1+1=2$$, so $$E$$ is at $$(3, 0)$$).
Now calculate the area of $$DGK$$ triangle from the coordinates of its vertices using the well-known formula: $$A=\frac12|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)|,$$ where $$x_i$$ and $$y_i$$ are the coordinates of $$i$$-th vertex ($$i=1,2,3$$).
Plugging $$x_1=0, y_1=1, x_2=1, y_2=2, x_3=s+3, y_3=s$$ gives $$A=\frac12|0(2-s)+1(s-1)+(s+3)(1-2)|=\frac12|s-1-s-3|=\frac12\times4=2.$$ Thus, the final result (which does not depends on $$s$$) is $$2\times20=40$$. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9796676466573412,
"lm_q1q2_score": 0.8660646946729094,
"lm_q2_score": 0.8840392939666336,
"openwebmath_perplexity": 126.29980772497434,
"openwebmath_score": 0.9412528872489929,
"tags": null,
"url": "https://puzzling.stackexchange.com/questions/109512/another-puzzle-with-area"
} |
Differential Entropy: Analytical Calculation and Sample-based Estimates
# Differential Entropy: Analytical Calculation and Sample-based Estimates
written by Ge Yang
THE GOAL$~~$ Compare the differential entropy computed analytically using the probability density function (PDF) with those estimated using a collection of samples.
# An Analytical Example
Consider a uniform distribution between $[0, 1/2]$
$p(x) = \begin{cases} 2&\text{ if } x \in [0, 1/2]\\ 0&\text{ otherwise. } \end{cases}$
The differential entropy is
\begin{aligned} H(u) &= - \int_0^{1/2} 2 * \log(2) \mathrm d x \\ &= - \log 2 \\ &= - 0.693 \end{aligned}
There are two ways to compute this numerically. The first is to use analytical calculation from the distribution. We do so by discretizing the probability distribution function into bins. We do so by normalizing the sequence of probabilities. This gives us $H(x) = -\log 2 = -0.69$:
def H_d(ps):
ps_norm = ps / ps.sum()
return - np.sum(np.log(ps) * ps_norm)
xs = np.linspace(0, 1 / 2, 1001)
ps = np.ones(1001) * 2
h_d = H_d(ps)
doc.print(f"analytical: {h_d:0.2f}")
analytical: -0.69
## Can We Compute This Via scipy entropy?
The scipy.stats.entropy function computes the Shannon entropy of a discrete distribution. The scipy entropy does not assume that the probability is normalized, so it normalizes the ps internally first. This means that in order to convert this to the differential entropy, we need to scale the result by the log sum.
h_d_analytic = stats.entropy(ps) - np.log(ps.sum())
doc.print(f"analytic entropy w/ scipy: {h_d_analytic:0.3f}")
analytic entropy w/ scipy: -0.693
This shows that we can get the discrete scipy.stats.entropy to agree with our differential entropy function H_d:ps -> R that is defined analytically.
# Estimating Entropy Using Samples from A Probability Density Function | {
"domain": "episodeyang.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.979667648438234,
"lm_q1q2_score": 0.86606468128213,
"lm_q2_score": 0.8840392786908831,
"openwebmath_perplexity": 3343.455063959063,
"openwebmath_score": 0.96881103515625,
"tags": null,
"url": "https://www.episodeyang.com/blog/2021/10-26/differential_entropy/"
} |
# Estimating Entropy Using Samples from A Probability Density Function
When we do not have access to the PDF, we can also estimate the entropy from a set of samples. We can do this using the differential_entropy function from the stats package in scipy:
from scipy import stats
samples = np.random.uniform(0, 1/2, 10001)
doc.print(f"verify the distribution: min: {samples.min():0.2f}, max: {samples.max():0.2f}")
verify the distribution: min: 0.00, max: 0.50
h_d_sample = stats.differential_entropy(samples)
doc.print(f"sample-based: {h_d_sample:0.3f}")
sample-based: -0.702
# Effect of Sampling On Differential Entropy
THE GOAL$~~$ vary the number of samples from the distribution, and compare the sample-based estimate against those computed from the analytical solution. This is done on the scipy documentation page 1.
Again, consider the uniform distribution from above.
def H_d(ps):
ps_norm = ps / ps.sum()
return - np.sum(np.log(ps) * ps_norm)
def get_H_d(N):
xs = np.linspace(0, 1 / 2, N)
ps = np.ones(N) * 2
h_d = H_d(ps)
doc.print(f"N={N} => h_d={h_d:0.2f}")
for n in [101, 201, 401, 801, 1601]:
get_H_d(n)
N=101 => h_d=-0.69
N=201 => h_d=-0.69
N=401 => h_d=-0.69
N=801 => h_d=-0.69
N=1601 => h_d=-0.69
The differential entropy does not depend on the number of points in the discretized distribution as long as the discretization is done properly.
# Effect of Change of Variable On Differential Entropy
THE GOAL$~~$ Investigate the effect of the change of variable, by varying the scale of the distribution and looking at the entropy. We can look at both the differential entropy analytically, and the sample-based estimates 1.
Now consider a more general case, a uniform distribution between $[a, b]$
$p(x) = \begin{cases} \frac 1 {\vert b - a \vert} & \text{if } x \in [a, b] \\ 0 & \text{otherwise. } \end{cases}$
The differential entropy is $\log \left(\vert b - a \vert\right)$. When a=0 and b=0.5, $H(x) = - \log 2$. | {
"domain": "episodeyang.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.979667648438234,
"lm_q1q2_score": 0.86606468128213,
"lm_q2_score": 0.8840392786908831,
"openwebmath_perplexity": 3343.455063959063,
"openwebmath_score": 0.96881103515625,
"tags": null,
"url": "https://www.episodeyang.com/blog/2021/10-26/differential_entropy/"
} |
We can verify this numerically. First let's define the entropy function
def H_d(ps):
ps_norm = ps / ps.sum()
return - np.sum(np.log(ps) * ps_norm)
We can plot the delta against $\log(b - a)$ -- it turned out to be zero across the range variants.
def get_H_d(a, b, N=201):
xs = np.linspace(a, b, N)
ps = np.ones(N) / (b - a)
h_d = H_d(ps)
delta = h_d - np.log(b - a)
doc.print(f"a={a}, b={b} => h_d={h_d:0.2f}, δ={delta}")
for b in [1/4, 1/2, 1, 2, 4]:
get_H_d(0, b)
a=0, b=0.25 => h_d=-1.39, δ=0.0
a=0, b=0.5 => h_d=-0.69, δ=0.0
a=0, b=1 => h_d=-0.00, δ=-0.0
a=0, b=2 => h_d=0.69, δ=0.0
a=0, b=4 => h_d=1.39, δ=0.0
## Using The Wrong Measure
What happens when you use the Shannon entropy on the samples, and treat the samples as the probabilities?
samples = np.random.uniform(0, 1/2, 10001)
doc.print(f"verify the distribution: min: {samples.min():0.2f}, max: {samples.max():0.2f}")
verify the distribution: min: 0.00, max: 0.50
h_d_wrong = H_d(samples)
doc.print(f"Shannon Entropy is (incorrectly): {h_d_wrong:0.3f}")
Shannon Entropy is (incorrectly): 1.195 | {
"domain": "episodeyang.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.979667648438234,
"lm_q1q2_score": 0.86606468128213,
"lm_q2_score": 0.8840392786908831,
"openwebmath_perplexity": 3343.455063959063,
"openwebmath_score": 0.96881103515625,
"tags": null,
"url": "https://www.episodeyang.com/blog/2021/10-26/differential_entropy/"
} |
Recent questions tagged gate2016 | {
"domain": "gateoverflow.in",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9796676496254958,
"lm_q1q2_score": 0.8660646808351999,
"lm_q2_score": 0.8840392771633078,
"openwebmath_perplexity": 793.7065183566209,
"openwebmath_score": 0.9886487722396851,
"tags": null,
"url": "https://ch.gateoverflow.in/tag/gate2016"
} |
Which one of the following is an iterative technique for solving a system of simultaneous linear algebraic equations? Gauss elimination Gauss-Jordan Gauss-Seidel $LU$ decomposition
The Laplace transform of $e^{at}\sin\left ( bt \right )$ is $\frac{b}{\left ( s-a \right )^{2}+b^{2}}$ $\frac{s-a}{\left ( s-a \right )^{2}+b^{2}}$ $\frac{s-a}{\left ( s-a \right )^{2}-b^{2}}$ $\frac{b}{\left ( s-a \right )^{2}-b^{2}}$
What are the modulus $\left ( r \right )$ and argument $\left ( \theta \right )$ of the complex number $3+4i$ ? $r=\sqrt{7}, \theta =tan^{-1}\left ( \frac{4}{3} \right )$ $r=\sqrt{7}, \theta =tan^{-1}\left ( \frac{3}{4} \right )$ $r={5}, \theta =tan^{-1}\left ( \frac{3}{4} \right )$ $r={5}, \theta =tan^{-1}\left ( \frac{4}{3} \right )$
A liquid mixture of ethanol and water is flowing as inlet stream $P$ into a stream splitter. It is split into two streams, $Q$ and $R$, as shown in the figure below. The flowrate of $P$, containing $30$ mass$\%$ of ethanol, is $100\:kg/h$. What is the ... additional specification$(s)$ required to determine the mass flowrates and composition (mass $\%$) of the two exit streams? $0$ $1$ $2$ $3$
The partial molar enthalpy (in $kJ$/mol) of species $1$ in a binary mixture is given by $\bar{h}_{1}=2 -60x_{2}^{2}+100x^{1}x_{2}^{2},$ where $x_{1}$ and $x_{2}$ are the mole fractions of species $1$ and $2$, respectively. The partial molar enthalpy (in $kJ$/mol, rounded off to the first decimal place) of species $1$ at infinite dilution is _________
For a flow through a smooth pipe, the Fanning friction factor $(f)$ is given by $f=mRe^{-0.2}$ in the turbulent flow regime, where $Re$ is the Reynolds number and $m$ is a constant. Water flowing through a section of this pipe with a velocity $1\:m/s$ results in a frictional ... the pressure drop across this section (in $kPa$), when the velocity of water is $\:2 m/s$? $11.5$ $20$ $34.8$ $40$ | {
"domain": "gateoverflow.in",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9796676496254958,
"lm_q1q2_score": 0.8660646808351999,
"lm_q2_score": 0.8840392771633078,
"openwebmath_perplexity": 793.7065183566209,
"openwebmath_score": 0.9886487722396851,
"tags": null,
"url": "https://ch.gateoverflow.in/tag/gate2016"
} |
In a cyclone separator used for separation of solid particles from a dust laden gas, the separation factor is defined as the ratio of the centrifugal force to the gravitational force acting on the particle. $S_{r}$ denotes the separation factor at a location (near the wall) ... $S_{r}$ depends on tangential velocity of the particle $S_{r}$ depends on the radial location $(r)$ of the particle
A vertical cylindrical vessel has a layer of kerosene (of density $800\: kg/m^{3}$) over a layer of water (of density $1000 \:kg/m^{3}$). $L$ - shaped glass tubes are connected to the column $30\: cm$ apart. The interface between the two layers lies between ... $cm$, rounded off to the first decimal place) of the interface from the point at which the lower $L$ - tube is connected is _________
A composite wall is made of four different materials of construction in the fashion shown below. The resistance (in $K/W$) of each sections of the wall is indicated in the diagram. The overall resistance (in $K/W$, rounded off to the first decimal place) of the composite wall, in the direction of heat flow, is ________
Steam at $100^{\circ}C$ is condensing on a vertical steel plate. The condensate flow is laminar. The average Nusselt numbers are $Nu_{1}$ and $Nu_{2}$, When the plate temperatures are $10^{\circ}C$ and $55^{\circ}C$, respectively. Assume the physical properties of the fluid and steel to ... -type condensation, what is the value of the ratio $\frac{Nu_{2}}{Nu_{1}}$ ? $0.5$ $0.84$ $1.19$ $1.41$
A binary liquid mixture of benzene and toluene contains $20$ mol$\%$ of benzene. At $350 \:K$ the vapour pressures of pure benzene and pure toluene are $92\: kPa$ and $35\: KPa$, respectively. The mixture follows Raoult's law. The equilibrium vapour phase mole fraction (rounded off to the second decimal place) of benzene in contact with this liquid mixture at $350\:K$ is ________ | {
"domain": "gateoverflow.in",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9796676496254958,
"lm_q1q2_score": 0.8660646808351999,
"lm_q2_score": 0.8840392771633078,
"openwebmath_perplexity": 793.7065183566209,
"openwebmath_score": 0.9886487722396851,
"tags": null,
"url": "https://ch.gateoverflow.in/tag/gate2016"
} |
Match the dimensionless numbers in $Group-I$ with the ratios in $Group-2$. $\begin{array}{cc}\\ &\text{Group-1} & \text{Group-2} \\ &\text{P. Biot number} & \text{ I.$\frac{buoyancy force}{viscous force}$} \\ &\text{ Q. Schmidt number} & \text{II.$\frac{internal thermal resistance of a solid}{boundary layer ... $Q-I$, $R-III$ $P-I$, $Q-III$, $R-II$ $P-III$, $Q-I$, $R-II$ $P-II$, $Q-III$, $R-I$
For what value of Lewis number, the wet-bulb temperature and adiabatic saturation temperature are nearly equal? $0.33$ $0.5$ $1$ $2$ | {
"domain": "gateoverflow.in",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9796676496254958,
"lm_q1q2_score": 0.8660646808351999,
"lm_q2_score": 0.8840392771633078,
"openwebmath_perplexity": 793.7065183566209,
"openwebmath_score": 0.9886487722396851,
"tags": null,
"url": "https://ch.gateoverflow.in/tag/gate2016"
} |
For a non-catalytic homogeneous reaction $A\rightarrow B$, the rate expression at $300\:K$ is $-r_{A}$\left (mol\; m^{-3} s^{-1}\right )=\frac{10C_{A}}{1+5C_{A}}$, where$C_{A}$is the concentration of$A$(in$mol\;/m^{3}$). Theoretically, the upper limit for the magnitude of the reaction rate ($-r_{A}\:in\:mol\:m^{-3}s^{-1}$, rounded off to the first decimal place) at$300\:K$is _________ 0 votes 0 answers The variations of the concentration$\left ( C_{A} ,C_{R}\:and\:C_{S}\right )$for three species$\left ( A,R\:and\:S \right )$with time, in an isothermal homogeneous batch reactor are shown in the figure below. Select the reaction scheme that correctly represents the above plot. The numbers in the reaction schemes shown below, represent the first order rate constants in unit of$s^{-1}$. 0 votes 0 answers Hydrogen iodide decomposes through the reaction$2HI\leftrightharpoons H_{_{2}}+I_{2}$. The value of the universal gas constant$R$is$8.314 \:J\:mol^{-1}K^{-1}$. The activation energy for the forward reaction is$184000\:J\:mol^{-1}$. The ratio (rounded off to the first decimal place) of the forward reaction rate at$600 \:K$to that at$550\:K$is ________ 0 votes 0 answers Match the instruments in$Group-I$with process variable in$Group-2$...$Q-I$,$R-IIIP-Il$,$Q-III$,$R-IP-III$,$Q-lI$,$R-IP-IIl$,$Q-I$,$R-Il$0 votes 0 answers What is the order of response exhibited by a$U$-tube manometer? Zero order First order Second order Third order 0 votes 0 answers A system exhibits inverse response for a unit step change in the input. Which one of the following statement must necessarily be satisfied? The transfer function of the system has at least one negative pole The transfer function of the system has at least one ... transfer function of the system has at least one negative zero The transfer function of the system has at least one positive zero 0 votes 0 answers Two design options for a distillation system are being compared based on the total annual cost. Information available is | {
"domain": "gateoverflow.in",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9796676496254958,
"lm_q1q2_score": 0.8660646808351999,
"lm_q2_score": 0.8840392771633078,
"openwebmath_perplexity": 793.7065183566209,
"openwebmath_score": 0.9886487722396851,
"tags": null,
"url": "https://ch.gateoverflow.in/tag/gate2016"
} |
a distillation system are being compared based on the total annual cost. Information available is as follows: ... above information, what is the total annual cost ($Rs$in lakhs/year) of the better option?$4042.492128$0 votes 0 answers Standard pipes of different schedule numbers and standard tubes of different$BWG$numbers are available in the market. For a pipe/tube of a given nominal diameter, which one the following statements is$TRUE$? Wall thickness increases with increase in both the ... the schedule number and the$BWG$number Neither the schedule number, nor the$BWG$number has any relation to wall thickness 0 votes 0 answers Terms used in engineering economics have standard definitions and interpretations. Which one of the following statements is$INCORRECT$? The profitability measure 'return on investment' does not consider the time value of money A cost index is an index value for ... different capacity Payback period is calculated based on the payback time for the sum of the fixed and the working capital investment 0 votes 0 answers India has no elemental sulphur deposits that can be economically exploited. In India, which one of the following industries produces elemental sulphur as a by-product? Coal carbonisation plants Petroleum refineries Paper and pulp industries Iron and steel making plants 0 votes 0 answers Two paper pulp plants$P$and$Q$...$P$and Plant$Q$both use the Kraft process Plant$P$uses Sulfite process Plant$P$uses Kraft process 0 votes 0 answers Match the industrial processes in$Group-I$with the catalyst materials in$Group-2$.$\begin{array}{cc}\\ \\ &\text{Group-1} & \text{Group-2} \\ &\text{P. Ethylene polymerisation} & \text{ I. Nickel} \\ &\text{ Q. Petroleum feedstock cracking} & \text{II. Vanadium pentoxide} \\ &\text{R. Oxidation of $...$R-III$,$S-IIP-I$,$Q-II$,$R-III$,$S-IVP-II$,$Q-III$,$R-IV$,$S-I$0 votes 0 answers A set of simultaneous linear algebraic equations is represented in a matrix form as shown below. ...$x_{3}$is ___________ | {
"domain": "gateoverflow.in",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9796676496254958,
"lm_q1q2_score": 0.8660646808351999,
"lm_q2_score": 0.8840392771633078,
"openwebmath_perplexity": 793.7065183566209,
"openwebmath_score": 0.9886487722396851,
"tags": null,
"url": "https://ch.gateoverflow.in/tag/gate2016"
} |
linear algebraic equations is represented in a matrix form as shown below. ...$x_{3}$is ___________ 0 votes 0 answers The model$y=mx^{2}$is to be fit to the data given below.$\begin{array}{|cl|cI|}\hline &{x} & {1} & {\sqrt{2}} & {\sqrt{3}} \\ \hline &{y} & {2} & {5} & {8} \\ \hline \end{array}$Using linear regression, the value (rounded off to the second decimal place) of$m$is _________ 0 votes 0 answers The Lagrange mean-value theorem is satisfied for$f\left ( x \right )=x^{3}+5$, in the interval$\left ( 1,4 \right )$at a value (rounded off to the second decimal place) of$x$equal to __________ 0 votes 0 answers Values of$f\left ( x \right )$in the interval$\left [ 0,4 \right ]$...$1$, the numerical approximation (rounded off to the second decimal place) of$\int_{0}^{4}f\left ( x \right )dx$is ______________ 0 votes 0 answers An ideal gas is adiabatically and irreversibly compressed from$3$bar and$300\:K$to$6$bar in a closed system. The work required for the irreversible compression is$1.5$times the work that is required for reversible compression from the same initial ... in$K\$, rounded off to the first decimal place) of the gas at the final state in the irreversible compression case is _______________ | {
"domain": "gateoverflow.in",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9796676496254958,
"lm_q1q2_score": 0.8660646808351999,
"lm_q2_score": 0.8840392771633078,
"openwebmath_perplexity": 793.7065183566209,
"openwebmath_score": 0.9886487722396851,
"tags": null,
"url": "https://ch.gateoverflow.in/tag/gate2016"
} |
Upper triangular is when all entries below the main diagonal are zero: An upper triangular matrix. When we multiply a matrix by a scalar value, then the process is known as scalar multiplication. If you add the m × n zero matrix to another m × n matrix A, you get A: In symbols, if 0 is a zero matrix and A is a matrix of the same size, then A + 0 = A and 0 + A = A A zero matrix is said to be an identity element for matrix addition. C. Unit matrix. Zero Matrix (Null Matrix) Zeros just everywhere: Zero matrix. Null Matrix A “null matrix” is one which has the value zero for all of its elements. It is a matrix with 0 in all its entries. A special kind of diagonal matrix in which all diagonal elements are the same is known as a scalar matrix. If we want to make A = a null matrix we need to multiply it by 0. [] is not a scalar and not a vector, but is a matrix and an array; something that is 0 x something or something by 0 is empty. The null space may also be treated as a subspace of the vector space of all n x 1 column matrices with matrix addition and scalar multiplication of a matrix as the two operations. Yes. Null matrix. C. is orthogonal. A square null matrix is also a diagonal matrix whose main diagonal elements are zero. An identity matrix is a scalar matrix with diagonal elements equal to one. Given a matrix and a scalar element k, our task is to find out the scalar product of that matrix. A square null matrix is also a diagonal . C. diagonal. For example, are null matrices of order 2x3 and 2x2. Given a matrix and a scalar element k, our task is to find out the scalar product of that matrix. Lower triangular is when all entries above the main diagonal are zero: A lower triangular matrix. Example 4.1 (Special Matrices) Some examples follow: 1. Note that the denominator of the fraction (just before the pivot's column vector) is the pivot itself (in this case “3”). Null Matrix Watch more videos at https://www.tutorialspoint.com/videotutorials/index.htm Lecture By: | {
"domain": "onepercentevent.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9796676490318648,
"lm_q1q2_score": 0.8660646773173749,
"lm_q2_score": 0.8840392741081575,
"openwebmath_perplexity": 455.83827433220875,
"openwebmath_score": 0.7774893641471863,
"tags": null,
"url": "http://onepercentevent.com/hiccups-linen-qhnvw/4715f0-is-null-matrix-a-scalar-matrix"
} |
Null Matrix Watch more videos at https://www.tutorialspoint.com/videotutorials/index.htm Lecture By: Er. By matrix-vector dot-product definition (a and u are vectors) $\begin\left\{bmatrix\right\} \begin\left\{array\right\}\left\{c\right\} a_1 \\ \hline \vdots \\ \hline a_n \\ \end\left\{array\right\} \end\left\{bmatrix\right\} * u = \left[a_1 * u, \dots, a_m * u\right]$ . Example 1. A scalar matrix has all main diagonal entries the same, with zero everywhere else: A scalar matrix . It is denoted by O. A unit matrix plays the role of the number 1 in numbers. 10. It is also a matrix and also an array; all scalars are also vectors, and all scalars are also matrix, and all scalars are also array There are several types of matrices, but the most commonly used are: Rows Matrix Columns Matrix Rectangular Matrix Square Matrix Diagonal Matrix Scalar Matrix Identity Matrix Triangular Matrix Null or Unit matrix and scalar matrix are special case of a diagonal matrix. It is never a scalar, but could be a vector if it is 0 x 1 or 1 x 0. The null matrix is the one in which all elements are zero. Z is a scalar matrix with lamda = 0. B. Scalar matrix. Null matrix or Zero-matrix : A matrix is said to be a null matrix or zero-matrix if each of its elements is zero. Is the scalar matrix is always a identity matrix? A square matrix m[][] will be diagonal matrix if and only if the elements of the except the main diagonal are zero. EASY. Two examples of a scalar matrix appear below. Let [([a.sub.ij]).sub. In other words, the square matrix A = [a ij] n×n is an identity matrix, if a ij = 1, when i = j and a ij = 0, when i ≠j. are all zero matrices. A diagonal matrix is a square matrix in which all the elements other than the principal diagonal elements are zero. Diagonal matrix. Then n x n matrix V = VX . D. has rank n. Solution: QUESTION: 9. A diagonal matrix, in which all diagonal elements are equal to same scalar, is called a scalar matrix. 11. Let us put into practice the knowledge | {
"domain": "onepercentevent.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9796676490318648,
"lm_q1q2_score": 0.8660646773173749,
"lm_q2_score": 0.8840392741081575,
"openwebmath_perplexity": 455.83827433220875,
"openwebmath_score": 0.7774893641471863,
"tags": null,
"url": "http://onepercentevent.com/hiccups-linen-qhnvw/4715f0-is-null-matrix-a-scalar-matrix"
} |
are equal to same scalar, is called a scalar matrix. 11. Let us put into practice the knowledge gained about the properties of matrix scalar multiplication and solve the next example exercises. Answer. Related Video. Any basis for the row space together with any basis for the null space gives a basis for . (i,j) [member of] AxB] be a scalar matrix with positive entries, and denote its columns by [[alpha].sub.j] = [([a.sub.ij]).sub.i[member of]A] and its rows by [[beta].sub.i] = [([a.sub.ij]).sub.j[member of]B]. It is a diagonal matrix with equal-valued elements along the diagonal. Properties of matrix addition & scalar multiplication. 6.5k VIEWS. Example : Identity Matrix. Question 3: Explain a scalar matrix? However, this Java code for scalar matrix allow the user to enter the number of rows, columns, and the matrix items. null. Null space of zero matrix. The types of matrices you have checked here are scalar matrix, unit and identity matrix, null or zero matrix, triangular matrix, with both options lower and upper triangular matrices. Program to check diagonal matrix and scalar matrix. Then diagonal matrix, rectangular matrix, row and column matrices. (i) A zero-matrix need not be a square matrix. Holder's inequality: some recent and unexpected applications. Program for scalar multiplication of a matrix. Triangular Matrix. A. Since the zero matrix is a small and concrete concept in itself which can be used through many of our lessons in linear algebra, we are now forced once more to enter into the topic of a later lesson: the null space of a matrix. Solution: QUESTION: 8. Rotation, coordinate scaling, and reflection. To show that the null space is indeed a vector space it is sufficient to show that , ∈ ⇒ + ∈ and ∈ ⇒ ∈ These are true due to the distributive law of matrices. However, in our case here, A 2 is not zero, and so we continue with Step 3. The dimension of the null space of a matrix is the nullity of the matrix. Step 3. In the special case when M is an m × | {
"domain": "onepercentevent.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9796676490318648,
"lm_q1q2_score": 0.8660646773173749,
"lm_q2_score": 0.8840392741081575,
"openwebmath_perplexity": 455.83827433220875,
"openwebmath_score": 0.7774893641471863,
"tags": null,
"url": "http://onepercentevent.com/hiccups-linen-qhnvw/4715f0-is-null-matrix-a-scalar-matrix"
} |
null space of a matrix is the nullity of the matrix. Step 3. In the special case when M is an m × m real square matrix, the matrices U and V * can be chosen to be real m × m matrices too. A diagonal matrix with all its main diagonal entries equal is a scalar matrix, that is, a scalar multiple λI of the identity matrix I. Here is the call graph for this function: Here is the caller graph for this function: m_matrix::m_matrix (const m_matrix & : that) D. Skew symmetric matrix. Matrix multiplication also known as matrix product . Google Classroom Facebook Twitter. When we add or subtract the 0 matrix of order m*n from any other matrix, it returns the same Matrix. We know that a matrix can be defined as an array of numbers. If a matrix A is symmetric as well as skew-symmetric, then A is a (A) Diagonal matrix (B) Null matrix asked Dec 6, 2019 in Trigonometry by Rozy ( 41.8k points) matrices We use the notation I p to denote a p×p identity matrix. 0, a matrix composed entirely of zeros, is called a null matrix. Just a note on separate Qs & As here. A submatrix of the given matrix can be obtained by deleting . Even a single number is stored as a matrix. 9. Let M be an arbitrary square matrix and Z be a zero matrix of the same dimension. A. has rank zero. It is a binary operation that produces a single matrix by taking two or more different matrices. A = 3: 0: 0: 3: B = 5: 0: 0: 0: 5: 0: 0: 0: 5: The identity matrix is also an example of a scalar matrix. Learn what a zero matrix is and how it relates to matrix addition, subtraction, and scalar multiplication. Symmetric. A matrix is a two-dimensional, rectangular array of data elements arranged in rows and columns. The most basic MATLAB® data structure is the matrix. We have to find whether the given square matrix is diagonal and scalar matrix or not, if it is diagonal and scalar matrix then print yes in the result. Examples: etc. Java Scalar Matrix Multiplication Program example 2. Answer: The scalar matrix is similar to a | {
"domain": "onepercentevent.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9796676490318648,
"lm_q1q2_score": 0.8660646773173749,
"lm_q2_score": 0.8840392741081575,
"openwebmath_perplexity": 455.83827433220875,
"openwebmath_score": 0.7774893641471863,
"tags": null,
"url": "http://onepercentevent.com/hiccups-linen-qhnvw/4715f0-is-null-matrix-a-scalar-matrix"
} |
etc. Java Scalar Matrix Multiplication Program example 2. Answer: The scalar matrix is similar to a square matrix. 6.5k SHARES. If M is a square matrix, is a scalar, and x is a vector satisfying then x is an eigenvector of M with corresponding eigenvalue . EDIT The question is this: Scalar multiplication is defined as B = A * s, where B and A are equally sized matrices (2D array of numbers, in this example let's use integers) and s is a scalar value. 263.1k SHARES. The elements can be numbers, logical values (true or false), dates and times, strings, or some other MATLAB data type. A square matrix A is symmetric if a ij = a ji for all i, j. The null matrix is also called the zero matrix. Then Z*M = Z = 0*M = 0 => Z = 0. If A 2 happens to be a null matrix, then the process terminates and the rank of A 1 is 1, which is then the largest subscript of a nonzero matrix. Diagonal matrix: A square matrix is said to be diagonal matrix if the elements of matrix except main diagonal are zero. The matrices » » » ¼ º « « « ¬ ª 0 0 0 0 0 Z and » » » ¼ º « « « ¬ ª 0 0 0 0 N are both null matrices. 263.1k VIEWS. The various types of matrices are row matrix, column matrix, null matrix, square matrix, diagonal matrix, upper triangular matrix, lower triangular matrix, symmetric matrix, and antisymmetric matrix. Select any nonzero element of A 2. Scalar Matrix. is said to be a scalar matrix if b ij = 0, when i ≠j b ij = k, when i =j, for some constant k. (vi) A square matrix in which elements in the diagonal are all 1 and rest are all zeroes is called an identity matrix. If M has n columns then rank(M)+nullity(M)=n. The product of any matrix by the scalar 0 is the null matrix. If equals (A) (B) zero matrix (C) a scalar quantity (D) identity matrix 1:36 35.6k LIKES. Scalar matrix and identity matrix 305.4k LIKES. Examples: Input : mat[][] = {{2, 3} {5, 4}} k = 5 Output : 10 . D. unit. Its effect on a vector is scalar multiplication by λ. It is a matrix with 0 in all its entries. Intro | {
"domain": "onepercentevent.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9796676490318648,
"lm_q1q2_score": 0.8660646773173749,
"lm_q2_score": 0.8840392741081575,
"openwebmath_perplexity": 455.83827433220875,
"openwebmath_score": 0.7774893641471863,
"tags": null,
"url": "http://onepercentevent.com/hiccups-linen-qhnvw/4715f0-is-null-matrix-a-scalar-matrix"
} |
effect on a vector is scalar multiplication by λ. It is a matrix with 0 in all its entries. Intro to zero matrices. B. has rank 1. Each element of A is multiplied to s, which is then stored in the corresponding element in matrix B. Is it true that the only matrix that is similar to a scalar matrix is itself Hot Network Questions Was the title "Prince of Wales" originally claimed for the English crown prince via a trick? X = [X 1, X 2, ... , X n] is an n — tuple non — zero vector. Ridhi Arora, Tutorials Point India Private Limited. View All. However a scalar matrix need not be a unit matrix. Email. A rectangular matrix 1234 5678 2. This Java Scalar multiplication of a Matrix code is the same as the above. In view of (7) like of the case of scalar matrix games the following theorem is proving. Example : Properties of Zero Matrix. Identity matrix is a scalar matrix in which all diagonal elements are 1. A 1 ×1 matrix is a scalar. A matrix is known as a zero or null matrix if all of its elements are zero. A scalar matrix is a special kind of diagonal matrix. Diagonal are zero, it returns the same, with zero everywhere else: a scalar value then... It is a matrix and a scalar matrix games the following theorem is proving matrix... Lower triangular matrix that matrix can be obtained by deleting n — tuple non — zero vector +nullity! We need to multiply it by 0 zero-matrix: a lower triangular is when all above! Dimension of the matrix items has n columns then rank ( M ) =n Java scalar multiplication solve! Note on separate Qs & is null matrix a scalar matrix here that a matrix is also called the zero matrix of the items... Columns then rank ( M ) +nullity ( M ) +nullity ( M ) +nullity M... Of numbers then rank ( M ) =n in which all diagonal elements are the is... Z be a vector if it is never a scalar, but could be a square in. Matrix and Z be a square matrix and scalar multiplication and solve the example! Principal diagonal elements are equal to one all elements are zero: an | {
"domain": "onepercentevent.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9796676490318648,
"lm_q1q2_score": 0.8660646773173749,
"lm_q2_score": 0.8840392741081575,
"openwebmath_perplexity": 455.83827433220875,
"openwebmath_score": 0.7774893641471863,
"tags": null,
"url": "http://onepercentevent.com/hiccups-linen-qhnvw/4715f0-is-null-matrix-a-scalar-matrix"
} |
and solve the example! Principal diagonal elements are equal to one all elements are zero: an upper triangular matrix an array data! Nullity of the same is known as a matrix case here, a matrix and a scalar games... To enter the number of rows, columns, and the matrix which is then stored in the element! & as here upper triangular is when all entries above the main diagonal zero... Solution: QUESTION: 9 unexpected applications a square matrix and a scalar value, then process... Matrix and a scalar is null matrix a scalar matrix are zero then diagonal matrix, in which all diagonal elements equal! Row space together with any basis for the following theorem is proving and columns zero-matrix: a matrix and scalar! = Z = 0 = > Z = 0 = > Z = 0 just... Be obtained by deleting or more different matrices if M has n columns then rank ( M ) =n rows. And unexpected applications a single matrix by the scalar matrix allow the user to enter the 1! Of data elements arranged in rows and columns continue with Step 3 upper triangular is when all above. Zeros just everywhere: zero matrix of the matrix a submatrix of the null space a... — tuple non — zero vector matrix are special case of a matrix and scalar matrix the diagonal matrix zero-matrix! And 2x2 the elements other than the principal diagonal elements are the same with. Number is stored as a matrix by a scalar matrix games the following theorem is.... Which has the value zero for all of its elements are 1 symmetric. Everywhere else: a lower triangular is when all entries above the main diagonal entries same. 1, x n ] is an n — tuple non — zero vector be defined as array. Notation i p to denote a p×p identity matrix is also a diagonal matrix if all of elements., it returns the same as the above to be diagonal matrix whose main diagonal entries the same.! The matrix a note on separate Qs & as here with 0 in all its entries example. Case of a diagonal matrix 1 or 1 x 0 are equal to.... A zero matrix is similar to a square matrix | {
"domain": "onepercentevent.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9796676490318648,
"lm_q1q2_score": 0.8660646773173749,
"lm_q2_score": 0.8840392741081575,
"openwebmath_perplexity": 455.83827433220875,
"openwebmath_score": 0.7774893641471863,
"tags": null,
"url": "http://onepercentevent.com/hiccups-linen-qhnvw/4715f0-is-null-matrix-a-scalar-matrix"
} |
Case of a diagonal matrix 1 or 1 x 0 are equal to.... A zero matrix is similar to a square matrix in which all diagonal elements equal one... Is always a identity matrix 1:36 35.6k LIKES and Z be a unit matrix plays the role the. But could be a zero or null matrix and how it relates to addition... Of that matrix following theorem is proving we use the notation i p to denote a p×p identity?. By taking two or more different matrices that produces a single number is as... Recent and unexpected applications arbitrary square matrix a zero-matrix need not be a null matrix need... Of Zeros, is called a null matrix or zero-matrix: a square matrix a “ null is. 1 in numbers or subtract the 0 matrix of order 2x3 and 2x2 and solve the next example exercises solve... One which has the value zero for all i, j ( null matrix Watch more videos at:. & as here are null matrices of order M * n from any other,. In matrix B, x n ] is an n — tuple non — zero vector null. A ) ( B ) zero matrix ( null matrix ” is which! To a square matrix in which all diagonal elements are zero is null matrix a scalar matrix diagonal we multiply a matrix scalar! And Z be a null matrix we need to multiply it by 0 need to multiply by! Watch more videos at https: //www.tutorialspoint.com/videotutorials/index.htm Lecture by: Er we add or subtract the 0 of. Zero or null matrix is a matrix a lower triangular is when all entries below the diagonal... Watch more videos at https: //www.tutorialspoint.com/videotutorials/index.htm Lecture by: Er the role the... As a scalar matrix need not be a zero or null matrix ) Zeros just everywhere: matrix... Matrix in which all diagonal elements are 1 our task is to find out the scalar in! If all of its elements: Er even a single matrix by the scalar is null matrix a scalar matrix with 0 all... Quantity ( D ) identity matrix scalar matrix has all main diagonal are zero with zero everywhere else a... Square null matrix is a matrix is the one in which all diagonal elements are 1: | {
"domain": "onepercentevent.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9796676490318648,
"lm_q1q2_score": 0.8660646773173749,
"lm_q2_score": 0.8840392741081575,
"openwebmath_perplexity": 455.83827433220875,
"openwebmath_score": 0.7774893641471863,
"tags": null,
"url": "http://onepercentevent.com/hiccups-linen-qhnvw/4715f0-is-null-matrix-a-scalar-matrix"
} |
everywhere else a... Square null matrix is a matrix is the one in which all diagonal elements are 1: recent! = > Z = 0 2x3 and 2x2 element in matrix B is when all above. By deleting same dimension zero matrix an identity matrix is a scalar matrix are special case of matrix! Scalar product of that matrix data elements arranged in rows and columns unexpected applications 2,... x... Never a scalar element k, our task is to find out the scalar 0 is null... “ null matrix any matrix by taking two or more different matrices = 0, j or null matrix is... ) a zero-matrix need not be a zero matrix view of ( 7 ) of. Matrix items is proving a note on separate Qs & as here the 0... = > Z = 0 * M = Z = 0 = Z = 0,,. A binary operation that produces a single matrix by taking two or more different matrices matrix ) just!, j to one Step 3 n. Solution: QUESTION: 9 submatrix of the case of scalar matrix all... By taking two or more different matrices subtract the 0 matrix of the case of diagonal. A binary operation that produces a single number is stored as a zero matrix of order *. Elements arranged in rows and columns us put into practice the knowledge gained about the of... A zero matrix +nullity ( M ) +nullity ( M ) =n gained about the properties of scalar... A is null matrix a scalar matrix symmetric if a ij = a null matrix a “ null matrix Watch more videos at https //www.tutorialspoint.com/videotutorials/index.htm... We continue with Step 3 zero vector its elements to matrix addition, subtraction, and matrix... Matrix with 0 in all its entries ( special matrices ) Some examples:. N ] is an n — tuple non — zero vector a special kind of diagonal matrix also a matrix. Elements is zero is one which has the value zero for all i j! Zero vector Z is a scalar matrix with lamda = 0 * M = 0 a two-dimensional, rectangular,. Rectangular matrix, row and column matrices examples follow: 1 0 M! Role of the number 1 in numbers ( i ) a zero-matrix need not be a null is! By a scalar matrix are | {
"domain": "onepercentevent.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9796676490318648,
"lm_q1q2_score": 0.8660646773173749,
"lm_q2_score": 0.8840392741081575,
"openwebmath_perplexity": 455.83827433220875,
"openwebmath_score": 0.7774893641471863,
"tags": null,
"url": "http://onepercentevent.com/hiccups-linen-qhnvw/4715f0-is-null-matrix-a-scalar-matrix"
} |
M! Role of the number 1 in numbers ( i ) a zero-matrix need not be a null is! By a scalar matrix are special case of a matrix with diagonal elements are the same, with zero else. = > Z = 0 * M = is null matrix a scalar matrix space of a is symmetric if ij... Value, then the process is known as scalar multiplication and solve the next example exercises Z. Same as the above into practice the knowledge gained about the properties of matrix except main diagonal elements zero... Non — zero vector by: Er 0 in all its entries by deleting same the. Is similar to a square matrix in which all elements are the same as the above we to! Matrix addition, subtraction, and the matrix items single matrix by the scalar matrix in which all diagonal are. Multiply it by 0 ( a ) ( B ) zero matrix ( null ”! To a square matrix in which all diagonal elements are zero same dimension on... Or zero-matrix: a scalar matrix with 0 in all its entries are zero notation i to! Equals ( a ) ( B ) zero matrix is also called the zero matrix is all... 35.6K LIKES in view of ( 7 ) like of the given matrix can be obtained by deleting (! Zeros just everywhere: zero matrix is said to be diagonal matrix is always a identity matrix p×p identity.... Z = 0 which all diagonal elements are zero its effect on a vector scalar... Data elements arranged in rows and columns 2x3 and 2x2 with zero everywhere:! The main diagonal elements are is null matrix a scalar matrix: a square matrix properties of matrix except main diagonal zero... Are the same dimension the role of the case of a diagonal matrix: a element... 2,..., x n ] is an n — tuple non — zero vector matrix... Be an arbitrary square matrix is a scalar matrix are 1 any basis for the null gives... Question: 9 a ij = a ji for all i, j in rows columns. Lamda = 0 * M = Z = 0 Z = 0 = > Z = 0 = Z. Null space gives a basis for subtract the 0 matrix of the case a... Case here, a matrix and Z be a zero matrix is the nullity of the given matrix can defined! For example, | {
"domain": "onepercentevent.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9796676490318648,
"lm_q1q2_score": 0.8660646773173749,
"lm_q2_score": 0.8840392741081575,
"openwebmath_perplexity": 455.83827433220875,
"openwebmath_score": 0.7774893641471863,
"tags": null,
"url": "http://onepercentevent.com/hiccups-linen-qhnvw/4715f0-is-null-matrix-a-scalar-matrix"
} |
here, a matrix and Z be a zero matrix is the nullity of the given matrix can defined! For example, are null matrices of order M * n from other. | {
"domain": "onepercentevent.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9796676490318648,
"lm_q1q2_score": 0.8660646773173749,
"lm_q2_score": 0.8840392741081575,
"openwebmath_perplexity": 455.83827433220875,
"openwebmath_score": 0.7774893641471863,
"tags": null,
"url": "http://onepercentevent.com/hiccups-linen-qhnvw/4715f0-is-null-matrix-a-scalar-matrix"
} |
# Probability a sample mean will fall in a range
1. Nov 21, 2015
### toothpaste666
1. The problem statement, all variables and given/known data
A random sample of size n = 81 is taken from an infinite population with the mean μ = 128 and the standard deviation σ = 6.3. With what probability can we assert that the value we obtain for the sample mean X will fall between 126.6 and 129.4?
3. The attempt at a solution
z = (x-μ)/(σ/sqrt(n))
so we have
z = (126.6-128)/(6.3/9) = -2 and z = (129.4-128)/(6.3/9) = 2
so the probability it will fall in the range is
F(2) - F(-2) = .9772 - .0228 = .9544
is this correct?
2. Nov 21, 2015
### Orodruin
Staff Emeritus
This depends on the actual distribution in the population. You can only do what you did if this distribution is assumed to be Gaussian.
3. Nov 21, 2015
### toothpaste666
Gaussian means "normal" right? I am confused a bit about that. In my book they seem to use "z" for the test statistic and use "t" when the population is known to be normal. From what I can tell they are the same thing except that with z you use the standard normal table and with t you use a different table with a certain amount of degrees of freedom. I don't think I fully get it.
4. Nov 21, 2015
### Ray Vickson
I do not actually believe you; I think you are mis-reading your book (although, to be honest, I am making this judgement sight-unseen). Typically, for an independent random sample from an underlying normal (=Gaussian) distribution with mean $\mu$ and variance $\sigma^2$: (1) we use $z$ and normal tables when we KNOW the value of $\sigma$; but (2) use $t$ and t-tables when we do not know $\sigma$, but have estimated it from the sample data itself. | {
"domain": "physicsforums.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9711290963960278,
"lm_q1q2_score": 0.8660636540140252,
"lm_q2_score": 0.891811044719067,
"openwebmath_perplexity": 621.4102277465222,
"openwebmath_score": 0.7689906358718872,
"tags": null,
"url": "https://www.physicsforums.com/threads/probability-a-sample-mean-will-fall-in-a-range.844276/"
} |
In case (2), we estimate
$$\text{estimator of }\: \sigma^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2$$
where the sample values are $x_1, x_2, \ldots, x_n$ and $\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i$ is the sample mean. In that case the jargon is that there are $n-1$ "degrees of freedom".
In the limit as $n \to \infty$ the t-distribution with (n-1) degrees of freedom becomes the standard normal, so using $z$ is like having infinitely many degrees of freedom.
5. Nov 21, 2015
### toothpaste666
so for either of the two statistics to work, the distribution must be normal?
6. Nov 21, 2015
### Ray Vickson
Theoretically, yes, but for a large sample-size, using the "normal" results give a "reasonably accurate" approximation. This is based on the so-called Central Limit Theorem; see, eg.,
https://en.wikipedia.org/wiki/Central_limit_theorem
or http://davidmlane.com/hyperstat/A14043.html
or http://www.statisticalengineering.com/central_limit_theorem.htm .
For a "reasonable" non-normal underlying distribution, a sample size of n = 81 is likely large enough that normal-based estimates will be informative, if not absolutely accurate.
7. Nov 21, 2015
### toothpaste666
ahh ok what my book actually says is use z for samples of n>30 with σ known and if σ is not known replace σ with s and if the sample is n<30 And the population is normal use t. so since my sample is large enough, my solution to this problem should be close enough?
8. Nov 21, 2015 | {
"domain": "physicsforums.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9711290963960278,
"lm_q1q2_score": 0.8660636540140252,
"lm_q2_score": 0.891811044719067,
"openwebmath_perplexity": 621.4102277465222,
"openwebmath_score": 0.7689906358718872,
"tags": null,
"url": "https://www.physicsforums.com/threads/probability-a-sample-mean-will-fall-in-a-range.844276/"
} |
# Double Sum Involving Condition
I would like to compute the dimensions of some small free nilpotent Lie algebras. However, I am totally new to this and I could not figure out how to write the double sum which gives the dimension of the free nilpotent Lie algebra $L(c,n)$ of step $c$ with $n$ generators.
Long story short, I would like to learn how I can write the sum
$$\dim L(c,n) = \sum_{m=1}^{c}\frac{1}{m}\sum_{d \mid m}\mu(d)n^{m/d}$$
where $\mu$ is the Möbius function, in Mathematica.
• Maybe diml[c_, n_] := Block[{j}, Sum[DirichletConvolve[MoebiusMu[j], n^j, j, m]/m, {m, c}]]? If this does what you want, I'll write an answer. – J. M. will be back soon Feb 18 '16 at 6:32
• @J.M. Hello. I tried this with some known values and it produced the correct values. I am sorry, since I do not know much about Mathematica this is the only way of me knowing that it works. Thank you! So please go ahead and write an answer, – Can Hatipoglu Feb 18 '16 at 7:11
• No need to apologize! You are fortunate here that Mathematica has the necessary stuff for the computations you want to do. Give me a few, and I'll write something up. – J. M. will be back soon Feb 18 '16 at 7:14
• FWIW, your inner sum equals JordanTotient[k, n] but unfortunately that function is not in Mathematica. – Chip Hurst Jun 2 '16 at 18:10
## A story of incremental improvement
Let's look at the OP's original expression again, for reference:
$$\sum_{m=1}^{c}\frac{1}{m}\sum_{d \mid m}\mu(d)n^{m/d}$$
Most people here are familiar with Sum[], and would not have much trouble translating the outer summation into Mathematica syntax. The inner part,
$$\sum_{d \mid m}\mu(d)n^{m/d}$$
is not terribly familiar to those who do not do much number theory. What this basically is saying is that the terms of the sum are indexed over the divisors of $m$. Appropriately enough, Mathematica does have a Divisors[] function. The inner sum can thus be written as
Sum[MoebiusMu[d] n^(m/d), {d, Divisors[m]}] | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.971129092218133,
"lm_q1q2_score": 0.866063649590024,
"lm_q2_score": 0.8918110440002044,
"openwebmath_perplexity": 1254.8681512725916,
"openwebmath_score": 0.5306399464607239,
"tags": null,
"url": "https://mathematica.stackexchange.com/questions/107652/double-sum-involving-condition"
} |
Sum[MoebiusMu[d] n^(m/d), {d, Divisors[m]}]
Summing over the divisors of a number, however, is so common a number-theoretic operation that Mathematica has seen it fit to provide a DivisorSum[] function. Thus, the inner sum can also be written as
DivisorSum[m, MoebiusMu[#] n^(m/#) &]
The story does not end here. The operation
$$\sum_{d \mid m}f(d)g(m/d)$$
frequently turns up in number-theoretic and other contexts, that it has been given a special name: Dirichlet convolution. Mathematica, fortunately, also provides a function for evaluating convolutions, called DirichletConvolve[]. Thus, the inner sum is most compactly expressed as
DirichletConvolve[MoebiusMu[j], n^j, j, m]
The function in the OP can now be implemented like so:
diml[c_Integer?Positive, n_Integer?Positive] := Block[{j},
Sum[DirichletConvolve[MoebiusMu[j], n^j, j, m]/m, {m, c}]]
or, using formal symbols as suggested by The Doctor,
diml[c_Integer?Positive, n_Integer?Positive] :=
Sum[DirichletConvolve[MoebiusMu[\[FormalJ]], n^\[FormalJ], \[FormalJ], m]/m, {m, c}]
(They look messy here on SE, but should look nice when pasted into Mathematica.)
Here is the function in action:
Table[diml[c, n], {c, 5}, {n, 5}]
{{1, 2, 3, 4, 5}, {1, 3, 6, 10, 15}, {1, 5, 14, 30, 55},
{1, 8, 32, 90, 205}, {1, 14, 80, 294, 829}}
• Thank you for introducing me to DirichletConvolve and for this instructive answer . This is my vote :) – ubpdqn Feb 18 '16 at 8:55
Please see J.M.'s answer (in comments) using DirichletConvolve. I would vote for this as an answer.
l[c_, n_] :=Module[{r = Range[c]},
Total[(1/r) Total /@MapThread[MoebiusMu@#1 n^(#2/#1) & ,{Divisors /@ r, r}]]]
I will happily delete and vote for JM answer.
I like J.M.'s answer and voted for it. However, it's hard to format comments, so I've posted this as an answer instead. Rather than using Block[{j},...], you can use formal symbols instead, used to represent a formal parameter that will never be assigned a value. Moreover, if you enter | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.971129092218133,
"lm_q1q2_score": 0.866063649590024,
"lm_q2_score": 0.8918110440002044,
"openwebmath_perplexity": 1254.8681512725916,
"openwebmath_score": 0.5306399464607239,
"tags": null,
"url": "https://mathematica.stackexchange.com/questions/107652/double-sum-involving-condition"
} |
DivisorSum[m, Function[d, MoebiusMu[d] n^(m/d)]] // TraditionalForm
you will see that Mathematica uses formal symbols automatically. Also, in many cases, I prefer to use the "map notation" for Function instead of # and &. For example,
is executable code for this function.
Finally, in TraditionalForm notation, the sum can be expressed as
which corresponds nicely to the original sum posted in the question. However, using DirichletConvolve should be more efficient. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.971129092218133,
"lm_q1q2_score": 0.866063649590024,
"lm_q2_score": 0.8918110440002044,
"openwebmath_perplexity": 1254.8681512725916,
"openwebmath_score": 0.5306399464607239,
"tags": null,
"url": "https://mathematica.stackexchange.com/questions/107652/double-sum-involving-condition"
} |
# Give an example of a linear transformation whose kernel is the line spanned by vector: $\begin{bmatrix} -1 & 1 & 2\end{bmatrix}^T$
I know that a linear transformation could be a projection onto the plane with normal vector $\begin{bmatrix} -1 & 1 & 2\end{bmatrix}^T$, but finding the projection would be too difficult.
I could easily think up a matrix where multiplied by $\begin{bmatrix} -1 & 1 & 2\end{bmatrix}^T = 0$, but I'm not sure on how to choose a matrix where $\begin{bmatrix} -1 & 1 & 2\end{bmatrix}^T$ is the only element of the kernel.
Also, can you please explain this Hint: "to describe a subset as a kernel means to describe it as an intersection of planes?"
• it's not too hard to cook up a matrix with rank 2 that kills this vector – qbert Jan 31 '18 at 4:53
• @qbert Yeah but how can I ensure nothing else send the matrix to 0? – Goldname Jan 31 '18 at 4:59
• "cook up" a $3\times 3$ matrix, making sure that (-1,1,2) is in the kernel and then show that the rank of this matrix is 2. And you know that rank + nulity = 3, so $(-1,1,2)$ must span the kernel. – Doug M Jan 31 '18 at 5:00
• if it has rank 2, it's kernel is rank 1. If it kills your given vector, then that spans the kernel – qbert Jan 31 '18 at 5:00
• ah I see, I forgot about that theorem, thanks – Goldname Jan 31 '18 at 5:18
Guide:
• $$2(-1) + 0(1) +1(2)=0$$
• $$0(01) + (-2)(1)+1(2)=0$$
• Verify that $\begin{bmatrix} 2 & 0 & 1 \end{bmatrix}$ and $\begin{bmatrix} 0 & -2 & 1 \end{bmatrix}$ are linearly independent.
• Now think of someway to form your matrix and prove that span of $\{\begin{bmatrix} -1 & 1 & 2 \end{bmatrix}^T\}$ is a basis to the kernel.
As for the explanation of the hint: To describe $x$ which satisfy $Ax=0$ where $a_i^T$ are the $i$-th row means $x$ is in the intersection of $\{ a_i^Tx = 0 : i \in \{ 1, \ldots, m\}\}$. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9711290880402379,
"lm_q1q2_score": 0.8660636486565645,
"lm_q2_score": 0.8918110468756548,
"openwebmath_perplexity": 173.12634429934533,
"openwebmath_score": 0.8747907280921936,
"tags": null,
"url": "https://math.stackexchange.com/questions/2629195/give-an-example-of-a-linear-transformation-whose-kernel-is-the-line-spanned-by-v"
} |
• I don't understand what your equations are supposed to mean – Goldname Jan 31 '18 at 4:59
• I can easily think of a matrix, but I don't understand what is required so that [-1 1 2] is the only element in the kernel – Goldname Jan 31 '18 at 4:59
• $[-1, 1, 2]$ is not the only element, any multiple of $[-1, 1, 2]$ is in the kernel. – Siong Thye Goh Jan 31 '18 at 5:01
• yeah that's what I meant – Goldname Jan 31 '18 at 5:03
• rank-nullity theorem is useful. geometrically, two non-parallel 3D plane intersect at a line. – Siong Thye Goh Jan 31 '18 at 5:03
Expanding upon the comment: Choosing the following matrix works $$\begin{bmatrix} 1&-1&1\\ 2&0&1\\ 1&1&0 \end{bmatrix}$$ It is clear that $(-1,1,2)$ is in the kernel. The dimension of the image is $2$, so the dimension of the kernel is just $1$, by rank nullity. So the only vectors that map to $0$ under the matrix are your vector and multiples of it.
Pick any two vectors that together with $[-1,1,2]^T$ form an ordered basis of $\mathbb R^3$. Relative to this basis, the matrix $\operatorname{diag}(1,1,0)$ represents a transformation with the required properties. Apply a change of basis to it to get a matrix relative to the standard basis. For that matter, since a linear transformation is completely determined by its action on a basis, the description $T(v_1)=v_1$, $T(v_2)=v_2$, $T([-1,1,2]^T)=0$, where $v_1$ and $v_2$ are the two vectors you chose, is a complete description of the transformation, but that’s likely not the answer that whoever gave you this problem is looking for.
As far as the hint goes, the kernel is a line through the origin, which can be described as the intersection of a pair of planes. The normals to those planes span the orthogonal complement of the line, so you can use them to construct the above basis. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9711290880402379,
"lm_q1q2_score": 0.8660636486565645,
"lm_q2_score": 0.8918110468756548,
"openwebmath_perplexity": 173.12634429934533,
"openwebmath_score": 0.8747907280921936,
"tags": null,
"url": "https://math.stackexchange.com/questions/2629195/give-an-example-of-a-linear-transformation-whose-kernel-is-the-line-spanned-by-v"
} |
• How did you conclude that "the matrix $diag(1,1,0)$ represents a transformation with the required properties"? Thanks – gbox Jan 31 '18 at 9:53
• @gbox The columns of a transformation matrix are the images of the basis vectors. In fact, for any linear transformation one can find bases for the domain and codomain such that the matrix has the form $\tiny{\left[\begin{array}{c|c}I&0\\\hline0&0\end{array}\right]}$. – amd Jan 31 '18 at 19:39
• @gbox BTW, $\operatorname{diag}(1,1,0)$ gives you a projection. If you instead choose some other pair of linearly independent vectors for the first two columns, you’ll get a different type of rank-2 transformation. – amd Jan 31 '18 at 19:47 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9711290880402379,
"lm_q1q2_score": 0.8660636486565645,
"lm_q2_score": 0.8918110468756548,
"openwebmath_perplexity": 173.12634429934533,
"openwebmath_score": 0.8747907280921936,
"tags": null,
"url": "https://math.stackexchange.com/questions/2629195/give-an-example-of-a-linear-transformation-whose-kernel-is-the-line-spanned-by-v"
} |
# Is there a name for the curve traced by the midpoint of two moving points on a circle with different speed?
I'm curious about what I did in Geogebra while I was experimenting with animation. Here it is:
1. Construct a circle with radius $$r$$
2. Define two points $$A(r\cos a\theta , r \sin a\theta)$$ and $$B(r\cos b\theta, r \sin b\theta)$$. Let $$(r,0)$$ be the initial position of both $$A$$ and $$B$$.
3. Get the midpoint of $$AB$$. Name this point $$M$$.
4. Change $$\theta$$ continuously starting from $$0$$ until $$A$$ and $$B$$ are both at $$(r,0)$$ again.
The midpoint, by doing the fourth step, seem to make some form of curve. I noticed that the curved traced by the midpoint does not change as long as $$a/b$$ does not change, even if $$a$$ and $$b$$ does so. Also, it doesn't matter if the values of $$a$$ and $$b$$ are swapped, but to avoid such ambiguities, we will avoid swapping of values and $$a$$ is always greater than $$b$$.
For the file, see this graph in Desmos.
For the angle, let $$p$$ be the numerator of $$a/b$$ in lowest terms. Then, we have $$0 \leq \theta \leq 2p\pi$$. (I don't know how to this in Desmos, by the way)
## Examples
For $$a = 2.1$$ and $$b = 1.4$$,
For $$a = 3$$ and $$b = 1$$,
And for $$a = 1.1$$ and $$b = 0.007$$ $$(0 \leq \theta \leq 98\pi)$$,
As for the midpoint, the coordinates is $$\left(\frac{r}{2}\left(\cos(a\theta) + \cos(b\theta)\right), \frac{r}{2}\left(\sin(a\theta) + \sin(b\theta)\right)\right)$$
This means that the parametric equation is \begin{align*} x(\theta) &= \frac{r}{2}\left(\cos(a\theta) + \cos(b\theta)\right) \\[10pt] y(\theta) &= \frac{r}{2}\left(\sin(a\theta) + \sin(b\theta)\right) \end{align*} Is there a name for these curves?
Update 1: It seems like this is somehow related to the mathematical basis of a spirograph. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.971129093889291,
"lm_q1q2_score": 0.866063644099297,
"lm_q2_score": 0.8918110368115781,
"openwebmath_perplexity": 281.0224501229029,
"openwebmath_score": 0.9765288233757019,
"tags": null,
"url": "https://math.stackexchange.com/questions/4192587/is-there-a-name-for-the-curve-traced-by-the-midpoint-of-two-moving-points-on-a-c"
} |
Update 1: It seems like this is somehow related to the mathematical basis of a spirograph.
... and therefore the trajectory equations take the form \begin{align*} x(t) &= R\left[(1-k)\cos t+lk\cos {\frac{1-k}{k}}t\right],\\y(t)&=R\left[(1-k)\sin t-lk\sin {\frac {1-k}{k}}t\right]\end{align*}
• It looks like what they call the evolute of a cardioid in this page (second figure) mathcurve.com/courbes2d.gb/largeur%20constante/… Jul 7 at 13:31
• Depending on a and b the curves can take many forms with multiple loops (b=1, a = 3) or spirals (a=2.3, b=2) so not cardioid, though one may be a special case.
– Paul
Jul 7 at 14:31
• This has the properties of several of the so-called transcendental curves, such as the sinusoidal spirals, Archimedean spiral, involute of a circle, and cochleoid. Curves are defined by equations, Cartesian and polar. You need to express this curve as an equation in order to define it, rather than a set of instructions. My go-to reference for plane curves is A Catalog of Special Plane Curves, by J. Dennis Lawrence, Dover Publications, 1972. Jul 9 at 14:59
• @CyeWaldman The parametric equation can be found before Update 1. Jul 9 at 16:24
• @soupless Thank you. From those equations, I would conclude that this is an unnamed curve. On a quick search the closest curve I can find to this is the hypotrochoid, described by $x=n\cos t+h\cos(n/b\cdot t)$ and $y=n\sin t-h\sin(n/b\cdot t)$. Jul 9 at 20:25
I am posting this since my updates that can answer my question should be posted as an answer, not as a part of the question.
We restrict the values of $$a$$ and $$b$$ to be integers where $$a$$ and $$b$$ are both nonzero. Then, the curve seems to represent an epitrochoid given by the parametric equation \begin{align*} x(t) &= m \cos t - h \cos\left(\frac{mt}{c}\right) \\ y(t) &= m \sin t - h \sin\left(\frac{mt}{c}\right). \end{align*} | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.971129093889291,
"lm_q1q2_score": 0.866063644099297,
"lm_q2_score": 0.8918110368115781,
"openwebmath_perplexity": 281.0224501229029,
"openwebmath_score": 0.9765288233757019,
"tags": null,
"url": "https://math.stackexchange.com/questions/4192587/is-there-a-name-for-the-curve-traced-by-the-midpoint-of-two-moving-points-on-a-c"
} |
where $$m = 1$$, $$h = 1$$, and $$c = \frac{a}{b}$$ for $$0 \leq t \leq 2a\pi$$.. For the parametric equation in the question, we let $$r = 2$$.
We can say that the two curves match if they are in the same magnitude and angle, and that they don't match if they don't have the same angle and magnitude. However, we'll say that they are a pseudomatch if they have the same magnitude but not having the same angle.
Now, for odd $$a$$, the two curve are a pseudomatch regardless of the value of $$b$$. For even $$a$$, there are two cases:
1. If $$a$$ is a power of $$2$$, then the curves are a pseudomatch if $$b$$ is a multiple of $$a$$. Otherwise, the curves are a match.
2. If $$a$$ is not a power of $$2$$, then the curves are a pseudomatch if $$b$$ is a multiple of the greatest power of $$2$$ that divides $$a$$. An example would be $$a = 28$$, where the curves are a pseudomatch if $$b$$ is a multiple of $$4$$.
As of the moment, I don't know how to prove that this works. I am just relying on graphs from the file. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.971129093889291,
"lm_q1q2_score": 0.866063644099297,
"lm_q2_score": 0.8918110368115781,
"openwebmath_perplexity": 281.0224501229029,
"openwebmath_score": 0.9765288233757019,
"tags": null,
"url": "https://math.stackexchange.com/questions/4192587/is-there-a-name-for-the-curve-traced-by-the-midpoint-of-two-moving-points-on-a-c"
} |
# Using functions to find indeterminate forms
The question is to use two functions f(x) and g(x) to show that ∞ - ∞ (infinity - infinity) is indeterminate.
I don't really know how to get started on this. I think I need to find two limits for f(x) and g(x) that are both infinity. But how to I show that subtracting them is indeterminate?
Edit: In the example we were shown, lim x --> 0 was used. But that was for the indeterminate form 0∞. I don't know if that applies here or not.
"Indeterminate" means "not having a unique value" as $x$ tends to whatever the problem says (I am guessing $x \rightarrow 0$ or $x \rightarrow \infty$, but you really should specify and not make the community you are asking for help guess). I will assume $x \rightarrow \infty$.
So, to show indeterminacy, I will exhibit three examples of the function pair $f(x), g(x)$ with different limiting behaviors of the difference $[f(x)-g(x)]$ as $x \rightarrow \infty$.
Example 1: $f(x) = g(x) = x$. Here $\lim_{x \rightarrow \infty}[f(x) - g(x)] = 0$.
Example 2: $f(x) = x^2, \; g(x) = x$.
Example 3: See Mark Fischler's answer.:)
• Thank you for the reply. The part that I don't understand is that when you use different values for f and g. Wouldn't you expect a different answer? I don't really understand how that makes it indeterminate. – user372991 Sep 28 '16 at 21:58
• Well consider a form that is not indeterminate: $1 / \infty$. In detail: if you have $$\lim_{x \rightarrow \infty} f(x) = \mbox{(some real cosntant)}, \quad \lim_{x \rightarrow \infty} g(x) = \infty,$$ then, no matter which $f, g$ you use, as long as they satisfy the above conditions, you will get $$\lim_{x \rightarrow} {f(x) \over g(x)} = 0.$$ In a high-level summary: The goal is, knowing the asymptotic behavior of $f, g$, to be able to ascertain the asymptotic behavior of an expression involving these two. When we cannot do that, the latter expression is indeterminate. – user8960 Sep 28 '16 at 23:18 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534314192864,
"lm_q1q2_score": 0.8660618666156011,
"lm_q2_score": 0.8824278757303678,
"openwebmath_perplexity": 176.3878151978045,
"openwebmath_score": 0.7851292490959167,
"tags": null,
"url": "https://math.stackexchange.com/questions/1945644/using-functions-to-find-indeterminate-forms"
} |
Consider $$f(x) = \csc^2x \\ g(x) = \cot^2 x \\ h(x) = 1/x^2$$ and look at the limit as $x\to 0$ of the expressions $f(x)-g(x)$ and $f(x)-h(x)$.
Both of these expressions are of the form $\infty - \infty$
But $$\lim_{x\to 0} (f(x) - g(x)) = 1 \\ \lim_{x\to 0} (f(x) - h(x)) = \frac13$$ This shows that $\infty - \infty$ could be $1$ or $\frac13$; for that matter it could be $0$ (consider $f(x)-f(x)$).
• If you wouldn't mind, could you explain how f(x) -g(x) = 1 and f(x) - h(x) = 1/3? I thought that they all would equal infinity? – user372991 Sep 28 '16 at 22:18
• @Jordan He is just giving you the answers for those limits. You can verify them with L'Hospital's Rule. It's a good exercise – imranfat Sep 28 '16 at 22:52
• On the other hand, the first one is particularly easy: Since $csc^2x - \cot^2x = 1$ for all values of angles where they are both finite, "obviously" the limit as $x\to$ any real value (or for that matter even any complex value) must be $1$. – Mark Fischler Oct 1 '16 at 21:20 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534314192864,
"lm_q1q2_score": 0.8660618666156011,
"lm_q2_score": 0.8824278757303678,
"openwebmath_perplexity": 176.3878151978045,
"openwebmath_score": 0.7851292490959167,
"tags": null,
"url": "https://math.stackexchange.com/questions/1945644/using-functions-to-find-indeterminate-forms"
} |
# how many subsets of $\{1,2,3,4,5,6,7\}$ has $6$ as largest?
Let $A =\{1,2,3,4,5,6,7\}$, The how many subsets have 6 as largest element?
My approach -
Total subsets are $2^7 = 128$. Then First I have to exclude the $1$ empty subset i.e. $\{\}$, $6$ sets with $1$ element except $\{6\}$, then two elements not containing $6$ as its element(I am not getting a way to count these element) and the $2$ subsets $\{6,7\},\{7,6\}$. Then I am stuck here as I am not getting the way to count these numbers. Any help?
• Well, none of the sets have 7 so this is the same as asking what are the subsets of B = {1.... 6} they all have 6 so all sets are equal to $C \cup \{6\}$ for some set C. C is a subset of $\{1,...., 5\}$ so we just need to find the subsets of {1,2...5} and add {6} to all of them. There are $2^5$ subset of {1,2....5} so there are the same number of subsets of {12,...5,6} that must have 6 which is the same number of subsets of {1,2,....7} that must have 6 and must not have 7. – fleablood Aug 25 '16 at 19:58
• 128 subsets. Half have 6 as an element. 1/4, contain both 6 and 7. So 1/4 have 6 and do not have 7. – Doug M Aug 25 '16 at 20:23
Remember how we determined that the number of subsets was 2 to the power of the number of elements.
We did that because for each element, $a$, either $a$ could be in a subset. So the total number of sets was the product of all the choices each of which was 2.
This is the same. Either 1 is in a subset or not. That 2 choices. Either 2 is in the subset or not. That's $2*2$ choice. Keep it up EXCEPT notice $6$ must be in the subset so that is only $1$ choice and $7$ must not be in the set so that's only one set.
So the number of sets is $2*2....*2*1*1$ which is what.
....
Or
.... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.986979508737192,
"lm_q1q2_score": 0.8660516197054005,
"lm_q2_score": 0.8774767986961403,
"openwebmath_perplexity": 292.80404298376527,
"openwebmath_score": 0.8369197845458984,
"tags": null,
"url": "https://math.stackexchange.com/questions/1903623/how-many-subsets-of-1-2-3-4-5-6-7-has-6-as-largest"
} |
So the number of sets is $2*2....*2*1*1$ which is what.
....
Or
....
Don't eliminate the sets one at a time. Remove ALL the subsets that do not have $6$ in them. How many do not have $6$. Remove ALL the subsets that do have $7$. How many is that? Then to avoid double counting add back the ones that had $6$ and didn't have $7$. How many is that?
....
Or
....
Figure 1/2 of the 128 have 6 and half do not. That only leaves half of them acceptable. Then half of those have 7 and half to not. That leaves only half of those acceptable.
....
Or
....
No set has $7$. So all the elements are taken from {$1,2,3,4,5,6$}.
All sets have $6$. So all elements that aren't $6$ are taken from {$1,2,3,4,5$} and $6$ is always added to it.
There are $2^5$ subsets of {$1,2,3,4,5$} and we are only taking those and sticking a $6$ into them. There are $2^5$ such sets.
• I like your way of dividing it up. That makes good sense! – John Aug 25 '16 at 20:16
• You have to be careful in assuming that your condition is exactly 1/2 though. In this case you can but if you were asked say How many sets have the numbers add up to an even number and contain the number 5 you can't assume half the sets add up to an even number and of those half of them contain 5. – fleablood Aug 25 '16 at 20:23
Hint:
Include $6$. How many ways are there to include zero through five members of $\{1,2,3,4,5\}$?
(Order doesn't matter with sets.) | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.986979508737192,
"lm_q1q2_score": 0.8660516197054005,
"lm_q2_score": 0.8774767986961403,
"openwebmath_perplexity": 292.80404298376527,
"openwebmath_score": 0.8369197845458984,
"tags": null,
"url": "https://math.stackexchange.com/questions/1903623/how-many-subsets-of-1-2-3-4-5-6-7-has-6-as-largest"
} |
(Order doesn't matter with sets.)
• okay now i got. There are total subsets = 2^6 = 64. – Brij Raj Kishore Aug 25 '16 at 19:58
• The element $6$ has to be there. The element $7$ can't be in there. But any, all, or none of $1$ through $5$ can be. In other words, if $S_5 = \{1,2,3,4,5\}$, then the union of $\{6\}$ and any subset of $S_5$ will have $6$ as the largest element. You figured out the number of subsets for a set of $7$ elements. How about for $5$ elements? – John Aug 25 '16 at 20:02
• Regarding your answer, not quite. It's not optional whether $6$ is in there or not. – John Aug 25 '16 at 20:03
• So the answer is 2^6 -1 - 5. Am i correct? But in option there are 4 options as - 32,31,64,128. – Brij Raj Kishore Aug 25 '16 at 20:05
• No set has 7. All sets have 6. So all sets are a subset of {1,2,3,4,5} unioned up with {6}. How many sets are there. There is {6} = {} + {6}; there is {1,4,5,6} = {1,4,5} + {6}. They are all C + {6} where $C \subset$ {1,2,3,4,5}. How many such sets are there. – fleablood Aug 25 '16 at 20:14
Think about it we need $6$ as an element but we can't have $7$ because $7$ is larger than $6$, we can have anything else.
What we're looking for: How many subsets have $6$ as an element but not $7$?
We must choose to include $6$, that gives $1$ choice. We may choose to to include $1$ or not, that gives $2$ choices. We may choose to include $2$ or not, that gives $2$ choices. We may choose to include $3$ or not, that gives $2$ choices. We may choose to include $4$ or not, that gives $2$ choices. We may choose to include $5$ or not, that gives $2$ choices. We must choose not to include $7$, that gives $1$ choice. By the multiplication principle the number subsets with $6$ but not $7$ is:
$$1•2•2•2•2•2•1=2^5$$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.986979508737192,
"lm_q1q2_score": 0.8660516197054005,
"lm_q2_score": 0.8774767986961403,
"openwebmath_perplexity": 292.80404298376527,
"openwebmath_score": 0.8369197845458984,
"tags": null,
"url": "https://math.stackexchange.com/questions/1903623/how-many-subsets-of-1-2-3-4-5-6-7-has-6-as-largest"
} |
# How to determine number with same amount of odd and even divisors
With given number N, how to determine first number after N with same amount of odd and even divisors? For example if we have N=1, then next number we are searching for is :
2
because divisors:
odd : 1
even : 2
I figured out that this special number can't be odd and obviously it can't be prime. I can't find any formula for this or do i have just compute it one by one and check if it's this special number ? Obviously 1 and number itself is divisors of this number. Cheers
-
Is this homework? How many powers of 2 can divide your special numbers? – user641 Jul 1 '11 at 11:27
We have holiday, i don't go to school :-) Is your question is a hint? – Spinach Jul 1 '11 at 11:52
Here's a hint. Try and pair up the even and odd factors in a natural way. You gave 2 = 1*2 as an example. How about factors of 2p, where p is an odd prime? – hardmath Jul 1 '11 at 12:05
Whenever you have two finite sets of the same size, you can think about whether there is a one-to-one correspondence (a bijection) between them. You might think whether you could apply this observation to the divisors of a number ... – Mark Bennet Jul 1 '11 at 12:07
To get some idea of what's going on, we do like other scientists do, we experiment.
Special numbers will be even, so we write down the number of odd divisors, even divisors, for the even numbers, starting with $2$. If a number turns out to be special, we put a $\ast$ in its row.
So we make a table, giving the number, how many odd divisors it has, how many even. Calculations are easy, but we must be very careful, since errors could lead us down the wrong path.
$2 \qquad 1 \qquad 1\quad\ast$
$4 \qquad 1 \qquad 2$
$6 \qquad 2 \qquad 2\quad\ast$
$8 \qquad 1 \qquad 3$
$10 \qquad 2 \qquad 2\quad\ast$
$12 \qquad 2 \qquad 4$
$14 \qquad 2 \qquad 2\quad\ast$
$16 \qquad 1 \qquad 4$
$18 \qquad 3 \qquad 3\quad\ast$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9869795083542037,
"lm_q1q2_score": 0.8660516083010532,
"lm_q2_score": 0.8774767874818408,
"openwebmath_perplexity": 268.2537258801534,
"openwebmath_score": 0.8541073203086853,
"tags": null,
"url": "http://math.stackexchange.com/questions/48876/how-to-determine-number-with-same-amount-of-odd-and-even-divisors"
} |
$14 \qquad 2 \qquad 2\quad\ast$
$16 \qquad 1 \qquad 4$
$18 \qquad 3 \qquad 3\quad\ast$
We could easily go on for a while. It is definitely not a waste of time, since it is useful to be well-acquainted with the structure of the smallish numbers that we bump into often.
A pattern seems to jump out: every second even number seems to be special. It looks as if "special" numbers are not all that special! It can be dangerous to jump to conclusions from data about small integers. But in this case, the conclusion turns out to be correct.
The special numbers, so far, all have the shape $2a$, where $a$ is an odd number. They are divisible by $2$ but not by $4$. The even numbers in our list that are not special are all divisible by $4$.
Now we try to prove that every number that is divisible by $2$ but not by $4$ is special, and that the others are not.
Take an odd number $b$, and look at the number $2b$. Think about the divisors of $2b$. If $k$ is an odd divisor of $2b$, then $2k$ is an even divisor of $2b$, and vice-versa.
If $k$ is an odd divisor of $2b$, call $2k$ the friend of $k$. Split the divisors of $2b$ into pairs of friends. For example, if $b=45$, we have the following pairs of friends.
$$(1,2)\qquad (3,6) \qquad(5,10)\qquad(9,18)\qquad(15,30) \qquad (45,90)$$
We have split the divisors of $2b$ into pairs of friends. Each pair has one odd number and one even number, so $2b$ has exactly as many odd divisors as even divisors.
Now let's show that no number divisible by $4$ can be special. The idea is that if a number is divisible by $4$, then it has "too many" even divisors. I will not write out the details, but you should. The idea goes as follows. Take a number $n$ that is divisible by $4$, like $36$ or $80$. Split the divisors of $n$ into teams. If $k$ is an odd divisor of $n$, put into the same team as $k$ the numbers $2k$, $4k$, and so on however far you can go.
Here are the teams for $n=36$. $$(1,2,4) \qquad (3,6,12)\qquad (9,18,36)$$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9869795083542037,
"lm_q1q2_score": 0.8660516083010532,
"lm_q2_score": 0.8774767874818408,
"openwebmath_perplexity": 268.2537258801534,
"openwebmath_score": 0.8541073203086853,
"tags": null,
"url": "http://math.stackexchange.com/questions/48876/how-to-determine-number-with-same-amount-of-odd-and-even-divisors"
} |
Here are the teams for $n=36$. $$(1,2,4) \qquad (3,6,12)\qquad (9,18,36)$$
Each team has more even numbers than odd numbers, so $n$ has more even divisors than odd divisors. That means $n$ can't be special.
Now let's get to your question: what is the first special number after $N$?
If $N$ is divisible by $4$, the first special number after $N$ is $N+2$. If $N$ is divisible by $2$ but not by $4$, the first special number after $N$ is $N+4$. If $N$ has remainder $1$ on division by $4$, the first special after $N$ is $N+1$, and if the remainder is $3$, the first special is $N+3$. These facts follow easily from what we have discovered about special numbers.
Formulas: We have been operating without formulas, just straight thinking. But I should mention a relevant formula. Let $n$ be an integer greater than $1$, and express $n$ as a product of powers of distinct primes. In symbols, let $$n=p_1^{a_1}p_2^{a_2} \cdots p_k^{a_k}$$ Then the number of divisors of $n$ is given by $$(a_1+1)(a_2+1) \cdots(a_k+1)$$
For example, $720=2^43^25^1$. The number of (positive) divisors of $n$ is $(4+1)(2+1)(1+1)$.
The formula that gives the number of divisors of $n$ is not hard to prove. Try to produce a proof! The formula could be adapted to give a count of the odd divisors of $n$, and of the even divisors. Then we could use these formulas to identify the special numbers. But formulas cannot do the thinking for you. So as a first approach, the way we tackled things is much better than trying to use a formula. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9869795083542037,
"lm_q1q2_score": 0.8660516083010532,
"lm_q2_score": 0.8774767874818408,
"openwebmath_perplexity": 268.2537258801534,
"openwebmath_score": 0.8541073203086853,
"tags": null,
"url": "http://math.stackexchange.com/questions/48876/how-to-determine-number-with-same-amount-of-odd-and-even-divisors"
} |
-
Thanks a lot ! Seems like experimenting is very useful, thanks for teaching me something :) And thanks for an so full answer. Chris – Spinach Jul 1 '11 at 14:20
+1; this answer embodies a very important lesson. A lot of questions here on math.SE could have been answered by the posters if they had decided to experiment first... – Qiaochu Yuan Jul 2 '11 at 4:11
@Qiaochu Yuan: A very common problem with students is that they do not really understand that mathematical questions, even those that seem quite abstract, are about concrete things. Unlike working mathematicians, students, even pretty good ones, are often naive formalists. – André Nicolas Jul 2 '11 at 4:28
I wrote a C++ application. My results are all the following numbers:
0, 2, 6, 10, 14, 18, 22, 26, 30, 34, ....
So, first zero, then two and then each time plus 4.
This means in general that if $n \equiv 2 (mod 4)$ is true, your number is one of the kind you're searching. Except from the first zero.
Here is the code of the application:
for (int i = 0; i < 1000; i += 2)
// Always +2 because of odd numbers don't have any even divisors
{
int even = 0;
int odd = 0;
for (int d = 1; d <= i; ++d)
{
if (i % d == 0)
{
if (d % 2 == 0)
{
even++;
} else
{
odd++;
}
}
}
if (even == odd)
{
printf("%d\n", i);
}
}
-
We know that there's always 1 divisor odd ( 1 ) and 1 divisor even for sure ( N ), so we need only to check to n/2 or even to sqrt(n) ( not sure if we won't miss some of divisors expect n/2 ). But nice application. – Spinach Jul 2 '11 at 14:01
For a given integer $n$, every divisor larger than $\sqrt{n}$ is paired with a divisor smaller than $\sqrt{n}$. Use this to figure out a general principle. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9869795083542037,
"lm_q1q2_score": 0.8660516083010532,
"lm_q2_score": 0.8774767874818408,
"openwebmath_perplexity": 268.2537258801534,
"openwebmath_score": 0.8541073203086853,
"tags": null,
"url": "http://math.stackexchange.com/questions/48876/how-to-determine-number-with-same-amount-of-odd-and-even-divisors"
} |
-
I don't think that's very much help in this case. There is a much simpler way, hinted at in the comments to the OP. – TonyK Jul 1 '11 at 13:46
@TonyK - I think this works as a pairing for this problem, though it involves making sure that you know that $n$ is not a square, so that there is a pairing - but since the number of distinct factors is known to be even, n can't be a square. – Mark Bennet Jul 1 '11 at 16:25
The pairing works of ncmathsadist works just fine, both for the proof that we have equality when $n \equiv 2 \pmod{4}$, and that there are more even divisors than odd when $n \equiv 0 \pmod{4}$. The pairing is in effect the same as mine when $n \equiv 2 \pmod 4$. But the mathematical description of the pairing is substantially better than mine, because the ncmathsadist pairing has a number of other uses. More detail would undoubtedly have been useful to the student. – André Nicolas Jul 2 '11 at 4:43
I didn't want to give away the whole thing. – ncmathsadist Jul 2 '11 at 13:25 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9869795083542037,
"lm_q1q2_score": 0.8660516083010532,
"lm_q2_score": 0.8774767874818408,
"openwebmath_perplexity": 268.2537258801534,
"openwebmath_score": 0.8541073203086853,
"tags": null,
"url": "http://math.stackexchange.com/questions/48876/how-to-determine-number-with-same-amount-of-odd-and-even-divisors"
} |
# Is it possible for an eigenspace to have more than one vector in its basis? What would that imply? Would every vector in the basis be an eigenvector?
So far I've been seeing that the vector that makes up a basis for the null space of the matrix $$A-Ix$$ (where x is an eigenvalue) is the eigenvector corresponding to the eigenvalue $$x$$. But I've never run into a situation where the basis had more than one vector in it.
Let's say the basis of the null space of $$A-2*I$$ was made up of two vectors. Would that mean those two (and only those two) vectors are the eigenvectors corresponding to the eigenvalue $$2$$ for the matrix $$A$$?
One thing of note is that, from what I understand, the identity transformation (in any space) has only one eigenvalue, $$1$$, but it has infinitely many eigenvectors (not sure if there is a vector space where this wouldn't be true, but it's true for $$R^n$$ at least). Not sure how this fact fits in here.
Just wanting to be sure I'm not misunderstanding something. I'm grateful for any help.
• Every nonzero vector in an eigenspace is an eigenvector. – amd Mar 9 at 20:10
Yes of course, you can have several vectors in the basis of an eigenspace.
First, when you have only one vector $$v$$ in a basis for a matrix $$A$$, with eigenvalue $$\mu$$, then any multiple of this vector is in the basis : $$\forall c \in \mathbb{R}, \ A(cv)=\mu(cv)$$
Then if you have two vectors in a basis, $$v$$ and $$w$$, any linear combination of these two vector will be an eigenvector for $$\mu$$: $$\left\{ \begin{array}{l} Av=\mu v\\ Aw=\mu w\\ \end{array} \right. \Rightarrow A(cv+dw) = \mu (cv+dw)$$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.974434786819155,
"lm_q1q2_score": 0.8660374818815622,
"lm_q2_score": 0.888758789809389,
"openwebmath_perplexity": 98.47559012698471,
"openwebmath_score": 0.9500262141227722,
"tags": null,
"url": "https://math.stackexchange.com/questions/3141173/is-it-possible-for-an-eigenspace-to-have-more-than-one-vector-in-its-basis-what"
} |
For exemple, let $$A=J-I$$ a matrix $$n\times n$$ of all 1, except 0 in the diagonal (this exemple comes from graph theory and the complete graph $$K_n$$). Then $$A$$ has eigenvalues $$n-1$$ and $$-1$$, and while the eigenspace associated with $$n-1$$ has dimension $$1$$ (there is only 1 vector in its basis), the eigenspace associated with the eigenvalue $$-1$$ has dimension $$n-1$$. Hence you can find an orthogonal basis of $$n-1$$ vectors. Any vector $$v$$ verifying $$Av=-v$$ can be written as a linear combination of the vectors in this basis.
You are understanding things just right.
For example, the $$3 \times 3$$ matrix $$\begin{bmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{bmatrix}$$ has two eigenvalues, $$2$$ and $$3$$.
Every nonzero vector on the $$x$$-axis is an eigenvector for eigenvalue $$2$$, and a basis for that eigenspace.
Every nonzero vector in the $$y$$-$$z$$ plane is an eigenvector for eigenvalue $$2$$. Any two linearly independent vectors in that plane form a basis of eigenvectors for that eigenvalue. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.974434786819155,
"lm_q1q2_score": 0.8660374818815622,
"lm_q2_score": 0.888758789809389,
"openwebmath_perplexity": 98.47559012698471,
"openwebmath_score": 0.9500262141227722,
"tags": null,
"url": "https://math.stackexchange.com/questions/3141173/is-it-possible-for-an-eigenspace-to-have-more-than-one-vector-in-its-basis-what"
} |
Question
# Let $$a$$ denote the element of the $${i^{th}}$$ row and $${j^{th}}$$ column in a $$3 \times 3$$ matrix and let $${a_{ij}} = \, - {a_{ji}}$$ for every i and j then this matrix is an -
A
Orthogonal matrix
B
singular matrix
C
matrix whose principal diagonal elements are all zero
D
skew-symmetric matrix
Solution
## The correct option is B skew-symmetric matrixIf $$a_{ij}$$ is the element of $$i^{th}$$ row and $$j^{th}$$ column and $$a_{ij}=-a_{ji}$$then lets assume$$A=\begin{bmatrix} { a }_{ 11 } & { a }_{ 12 } & { a }_{ 13 } \\ { a }_{ 21 } & { a }_{ 22 } & { a }_{ 23 } \\ { a }_{ 31 } & { a }_{ 32 } & { a }_{ 33 } \end{bmatrix}$$Where, $$a_{ij}=-a_{ji}$$$$a_{11}=-a_{11}\implies a_{11}=0$$, similarly $$a_{22}=a_{33}=0$$and $$a_{21}=-a_{12}, a_{31}=-a_{13}, a_{32}=-a_{23}$$Putting the values$$A=\begin{bmatrix} { 0 } & { a }_{ 12 } & { a }_{ 13 } \\ { -a }_{ 12 } & { 0 } & { a }_{ 23 } \\ { -a }_{ 13 } & { -a }_{ 23 } & { 0 } \end{bmatrix}$$$${ A }^{ T }=\begin{bmatrix} { 0 } & { -a }_{ 12 } & { -a }_{ 13 } \\ { a }_{ 12 } & { 0 } & { -a }_{ 23 } \\ { a }_{ 13 } & { a }_{ 23 } & {0 } \end{bmatrix}\\$$and$$-A=\begin{bmatrix} { 0 } & -{ a }_{ 12 } & -{ a }_{ 13 } \\ { a }_{ 12 } & { 0 } & -{ a }_{ 23 } \\ { a }_{ 13 } & { a }_{ 23 } & { 0 } \end{bmatrix}$$thus$$A^T=-A$$$$\therefore$$ The matrix is a skew-symmetric matrix.Mathematics
Suggest Corrections
0
Similar questions
View More
People also searched for
View More | {
"domain": "byjus.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9888419710536893,
"lm_q1q2_score": 0.8660149446618723,
"lm_q2_score": 0.8757870013740061,
"openwebmath_perplexity": 1213.7262986456699,
"openwebmath_score": 0.9499050974845886,
"tags": null,
"url": "https://byjus.com/question-answer/let-a-denote-the-element-of-the-i-th-row-and-j-th-column-in/"
} |
Need to prove the sequence $a_n=1+\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{n^2}$ converges
I need to prove that the sequence $a_n=1+\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{n^2}$ converges. I do not have to find the limit. I have tried to prove it by proving that the sequence is monotone and bounded, but I am having some trouble:
Monotonic:
The sequence seems to be monotone and increasing. This can be proved by induction: Claim that $a_n\leq a_{n+1}$
$$a_1=1\leq 1+\frac{1}{2^2}=a_2$$
Need to show that $a_{n+1}\leq a_{n+2}$
$$a_{n+1}=1+\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{n^2}+\frac{1}{(n+1)^2}\leq 1+\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{n^2}+\frac{1}{(n+1)^2}+\frac{1}{(n+2)^2}=a_{n+2}$$
Thus the sequence is monotone and increasing.
Boundedness:
Since the sequence is increasing it is bounded below by $a_1=1$.
Upper bound is where I am having trouble. All the examples I have dealt with in class have to do with decreasing functions, but I don’t know what my thinking process should be to find an upper bound.
Can anyone enlighten me as to how I should approach this, and can anyone confirm my work thus far? Also, although I prove this using monotonicity and boundedness, could I have approached this by showing the sequence was a Cauchy sequence?
Solutions Collecting From Web of "Need to prove the sequence $a_n=1+\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{n^2}$ converges"
Your work looks good so far. Here is a hint:
$$\frac{1}{n^2} \le \frac{1}{n(n-1)} = \frac{1}{n-1} – \frac{1}{n}$$
To elaborate, apply the hint to get:
$$\frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \cdots + \frac{1}{n^2} \le \left(\frac{1}{1} – \frac{1}{2}\right) + \left(\frac{1}{2} – \frac{1}{3}\right) + \left(\frac{1}{3} – \frac{1}{4}\right) + \cdots + \left(\frac{1}{n-1} – \frac{1}{n}\right)$$
Notice that we had to omit the term $1$ because the inequality in the hint is only applicable when $n > 1$. No problem; we will add it later. | {
"domain": "bootmath.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9888419703960399,
"lm_q1q2_score": 0.8660149440859115,
"lm_q2_score": 0.8757870013740061,
"openwebmath_perplexity": 209.7219452408486,
"openwebmath_score": 0.9254652857780457,
"tags": null,
"url": "http://bootmath.com/need-to-prove-the-sequence-a_n1frac122frac132cdotsfrac1n2-converges.html"
} |
Also notice that all terms on the right-hand side cancel out except for the first and last one. Thus:
$$\frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \cdots + \frac{1}{n^2} \le 1 – \frac{1}{n}$$
Add $1$ to both sides to get:
$$a_n \le 2 – \frac{1}{n} \le 2$$
It follows that $a_n$ is bounded from above and hence convergent.
It is worth noting that canceling behavior we saw here is called telescoping. Check out the wikipedia article for more examples.
Besides to Ayman’s neat answer, you may take $f(x)=\frac{1}{x^2}$ over $[1,+\infty)$ and see that $f'(x)=-2x^{-3}$ and then is decreasing over $[1,+\infty)$. $f(x)$ is also positive and continuous so you can use the integral test for $$\sum_{k=1}^{+\infty}\frac{1}{n^2}$$ to see the series is convergent. Now your $a_n$ is the $n-$th summation of this series.
You can show this geometrically too.
If you take a square, and divide the height into $\frac 12$, $\frac 14$, $\frac 18$, and so forth, doubling the denominator on each deal.
Now take the squares $\frac 12$ and $\frac 13$: these go onto the top shelf.
On the second shelf go $\frac 14$ to $\frac 17$. These fractions are all less than $\frac 14$, so fit onto the second shelf. Likewise, squares from $8$ to $15$ go onto the third shelf, each $\frac 1n$ is smaller than $\frac 18$, and so forth.
Therefore $\sum_{n=2}^{\infty}\frac 1n < 1$, and therefore the whole lot is less than two squares.
by the integral test :
$\int^\infty_1 \frac{1}{n^2} dn \le \sum_{i=1}^\infty \frac{1}{n^2}\le 1+\int^\infty_1 \frac{1}{n^2}dn$ .
you can compute the integral so the answer is :
$1 \le \sum_{i=1}^\infty \frac{1}{n^2}\le 2$ .
because : $\int^\infty_1 \frac{1}{n^2} dn =1$
Hint: Prove the the following holds for all $n$ by induction.
$$\sum_1^n \frac{1}{k^2} \le 2 – \frac{1}{n}.$$
Is it not uncommon that when proving some inequality by induction, you will first need to strengthen the hypothesis to get the induction to work. | {
"domain": "bootmath.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9888419703960399,
"lm_q1q2_score": 0.8660149440859115,
"lm_q2_score": 0.8757870013740061,
"openwebmath_perplexity": 209.7219452408486,
"openwebmath_score": 0.9254652857780457,
"tags": null,
"url": "http://bootmath.com/need-to-prove-the-sequence-a_n1frac122frac132cdotsfrac1n2-converges.html"
} |
You can use the same technique to bound other values of the zeta function. For example, try showing $\zeta(3)$ is bounded above by $\frac{3}{2}$.
(I am copying my answer from a duplicate question that was closed as a copy of this one since the induction approach is not available here.)
This here should work with $n \geq 1$ :
$$s_n = \sum\limits_{n=1}^\infty \frac{1}{n^2}= \frac{1}{1} + \frac{1}{4} +\frac{1}{9} + \frac{1}{16}+ …+ \frac{1}{n²}$$$$b_n = \sum\limits_{n=1}^\infty \frac{1}{2^{n-1}} = \frac{1}{1} + \frac{1}{2} +\frac{1}{4}+\frac{1}{8} + …+ \frac{1}{2^{n-1}}$$
$b_n$ is directly compared greater than $s_n$ :
$$s_n < b_n$$
and $b_n$ converges, because of its ratio test :
$$\frac{1}{2^{n-1+1}} / \frac{1}{2^{n-1}} = \frac{2^{n-1}}{2^{n}} = \frac{1}{2} < 1$$
Notice that $2k^2 \geq k(k+1) \implies \frac{1}{k^2} \leq \frac{2}{k(k+1)}$.
$$\sum_{k=1}^{\infty} \frac{2}{k(k+1)} = \frac{2}{1 \times 2} + \frac{2}{2 \times 3} + \frac{2}{3 \times 4} + \ldots$$
$$\sum_{k=1}^{\infty} \frac{2}{k(k+1)} = 2\Big(\, \Big(1 – \frac{1}{2}\Big) + \Big(\frac{1}{2} – \frac{1}{3} \Big) + \Big(\frac{1}{3} – \frac{1}{4} \Big) + \ldots \Big)$$
$$\sum_{k=1}^{\infty} \frac{2}{k(k+1)} = 2 (1) = 2$$.
Therefore $\sum_{k=1}^{\infty} \frac{1}{k^2} \leq 2$.
$a_n=\frac{1}{n^2}$ and because $a_n>0$ we have $|a_n|=a_n$
First: check the necessary condition
$$\lim_{n\to +\infty} na_n=\lim_{n\to +\infty}\frac{1}{n}=0$$
Second: check D’Alembert’s ratio test
$$\lim_{n\to +\infty}|\frac{a_{n+1}}{a_n}|=\lim_{n\to +\infty}\frac{a_{n+1}}{a_n}=\lim_{n\to +\infty}(\frac{n}{n+1})^2=1$$
Third: Because the answer of D’Alembert’s test is $1$, you should use Raabe’s test:
$$\lim_{n\to +\infty}n(1-|\frac{a_{n+1}}{a_n}|)=\lim_{n\to +\infty}n(1-\frac{a_{n+1}}{a_n})=\lim_{n\to +\infty}n(\frac{2n+1}{n^2+2n+1})=2\gt1$$
so the series is convergent | {
"domain": "bootmath.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9888419703960399,
"lm_q1q2_score": 0.8660149440859115,
"lm_q2_score": 0.8757870013740061,
"openwebmath_perplexity": 209.7219452408486,
"openwebmath_score": 0.9254652857780457,
"tags": null,
"url": "http://bootmath.com/need-to-prove-the-sequence-a_n1frac122frac132cdotsfrac1n2-converges.html"
} |
# Thread: Show that 64 divides 9^n + 56n + 63
1. ## Show that 64 divides 9^n + 56n + 63
Show that 64 divides 9^n + 56n + 63 for all positive integers n.
Based off of basic computation I can see that for numbers like n=1, n=2, etc this holds but how would I show it for all positive integers?
2. ## Re: Show that 64 divides 9^n + 56n + 63
are you familiar with induction?
$9 + 56 + 63 = 128 = 2 \cdot 64$
so this is true for $n=1$ and we would say $P_1 = True$
next, we assume that $P_n=True$ and we use this to show that $P_{n+1} = True$
here we'd do
\begin{align*} &9^{n+1} + 56(n+1) + 63 \\ = &~9^n\cdot 9 + 56n + 56 + 63 \\ = &~9^n+ 56n + 63 + 8\cdot 9^n + 56 \\ = &~64k + 8\cdot 9^n + 56 \text{; because }P_n=True \\ =&~64k + 8(7+9^n) \end{align*}
clearly $64k$ is divisible by $64$ so we need to show that $8(7+9^n)$ is divisible by $64$ or alternatively that
$(7+9^n)$ is divisible by $8$
Using the same method we are using to show the original statement
$n=1 \Rightarrow 7+9^n = 16 = 2\cdot 8$ so $\tilde{P}_1=True$
Now you show that $\tilde{P}_n \Rightarrow \tilde{P}_{n+1}$ for this and thus that $7+ 9^n$ is divisible by 8
and you will have shown that
$9^{n+1} + 56(n+1) + 63$ is divisible by $64$
and thus that for the original statement
$P_n \Rightarrow P_{n+1}$
and thus it is True for all $n\geq 1$
so see if you can prove that $\tilde{P}_n \Rightarrow \tilde{P}_{n+1}$ it's not hard
3. ## Re: Show that 64 divides 9^n + 56n + 63
You can also note:
$\displaystyle 9^n + 56n + 63 =$
$\displaystyle 9^n + 56n + (8n - 8n) + 63 + (1 - 1) =$
$\displaystyle 9^n + (56n + 8n) + (63 + 1) - 1 - 8n =$
$\displaystyle 9^n + (64n + 64) - 1 - 8n$
64 divides (64n + 64), so it remains to show that 64 also divides $\displaystyle \ \ 9^n - 1 - 8n$.
It has been shown or mentioned it's true for n = 1 in at least one prior post. Let's look at $\displaystyle \ \ n \ge 2$:
$\displaystyle 9^n - 1 - 8n =$
$\displaystyle (8 + 1)^n - 1 - 8n =$ | {
"domain": "mathhelpforum.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9850429129677614,
"lm_q1q2_score": 0.8659972927937779,
"lm_q2_score": 0.8791467675095292,
"openwebmath_perplexity": 375.1864141124909,
"openwebmath_score": 0.9954174160957336,
"tags": null,
"url": "http://mathhelpforum.com/discrete-math/274100-show-64-divides-9-n-56n-63-a.html"
} |
$\displaystyle 9^n - 1 - 8n =$
$\displaystyle (8 + 1)^n - 1 - 8n =$
$\displaystyle \bigg(8^n \ + \ n\cdot8^{n - 1} \ + \ \dfrac{n(n - 1)}{2}8^{n - 2} \ + \ . . . \ + \ \dfrac{n(n - 1)}{2}8^2 \ + \ n\cdot8^1 \ + \ 1\bigg) \ - \ 1 \ - \ 8n =$
$\displaystyle 8^n \ + \ n\cdot8^{n - 1} \ + \ \dfrac{n(n - 1)}{2}8^{n - 2} \ + \ . . . \ + \ \dfrac{n(n - 1)}{2}8^2 \ + \ 8n \ - \ 8n \ + \ 1 \ - \ 1 =$
$\displaystyle 8^n \ + \ n\cdot8^{n - 1} \ + \ \dfrac{n(n - 1)}{2}8^{n - 2} \ + \ . . . \ + \ \dfrac{n(n - 1)}{2}8^2$ . . . . . . . . . **
The fractions in front of the powers of eights are combinations and are thus positive integers. The smallest power of eight is equal to 64,
so this expression is also divisible by 64.
** . . . . If n = 2, there will only be one term here. If n = 3, there will only be two terms here. And so on.
4. ## Re: Show that 64 divides 9^n + 56n + 63
The induction step can be done as follows:
\displaystyle \begin{align*}9^{n+1} + 56(n+1) + 63 &= 9\cdot 9^n + 56n + 56 + 63 \\ &= 8( 9^n + 7) + (9^n + 56n + 63) \end{align*}
where the final parenthesis is a multiple of 64 by the inductive hypothesis. So all we need is for $\displaystyle 8$ to divide $\displaystyle (9^n + 7)$.
$\displaystyle 9^n + 7 \equiv 1^n - 1 \pmod{8} = 1-1 \pmod{8} = 0 \pmod{8}$
QED | {
"domain": "mathhelpforum.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9850429129677614,
"lm_q1q2_score": 0.8659972927937779,
"lm_q2_score": 0.8791467675095292,
"openwebmath_perplexity": 375.1864141124909,
"openwebmath_score": 0.9954174160957336,
"tags": null,
"url": "http://mathhelpforum.com/discrete-math/274100-show-64-divides-9-n-56n-63-a.html"
} |
# Proving an expression is composite
I am trying to prove that $n^4 + 4^n$ is composite if $n$ is an integer greater than 1. This is trivial for even $n$ since the expression will be even if $n$ is even.
This problem is given in a section where induction is introduced, but I am not quite sure how induction could be used to solve this problem. I have tried examining expansions of the expression at $n+2$ and $n$, but have found no success.
I would appreciate any hints on how to go about proving that the expression is not prime for odd integers greater than 1.
-
Write it as a difference of squares. – Adam Hughes Jul 1 '14 at 19:54
How? It's $n^4 + 4^n$ not $n^4 - 4^n$ – Mathmo123 Jul 1 '14 at 19:55
@Mathmo123: see my answer below. – Adam Hughes Jul 1 '14 at 20:03
$(n^2)^2+(2^n)^2=(n^2+2^n)^2-2^{n+1}n^2$. Since $n$ is odd...
-
Ah. Very neat!! – Mathmo123 Jul 1 '14 at 20:05
Originally, I was not sure how I should go about writing it as a difference of squares. Thanks for your help. – pidude Jul 1 '14 at 20:12
Hint: calculate this value explicitly for $n=1,3$ (or predict what will happen). Can you see any common factors? Can you prove that there is a number $m$ such that if $n$ is odd, then $m|(n^4 + 4^n)$?
Let me know if you need further hints.
-
I originally tried doing that but to no avail. Evaluated at 3 I obtain 145 which has factors of 5 and 29. At 5, the expression equals 1649, which has factors of 17 and 97. I will keep looking for a pattern and let you know if I need more hints. Thanks for your help. – pidude Jul 1 '14 at 19:59
Ah... using this method, there will be a difference between multiples of 5 and other odd numbers. You will find that for odd numbers that are not a multiple of 5, 5 will be a divisor – Mathmo123 Jul 1 '14 at 20:01
Ok, so I was able to prove that. Now I'm working numbers which are multiples of 5. – pidude Jul 1 '14 at 20:09 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9850429116504951,
"lm_q1q2_score": 0.8659972853975705,
"lm_q2_score": 0.879146761176671,
"openwebmath_perplexity": 439.9506829252601,
"openwebmath_score": 0.9091550707817078,
"tags": null,
"url": "http://math.stackexchange.com/questions/853615/proving-an-expression-is-composite"
} |
I am very impressive with Adam's solution. There is very neat. So, I beg for a chance to write the full description about the proof step-by-step.
• We can transform $n^4+4^n$ to $(n^2+2^n)^2-2^{n+1}n^2$ as Adam's suggestion by
1. $n^{(2^2)}+(2^2)^n = (n^2)^2+(2^n)^2$ associative law
2. Now, we mention $(a+b)^2 = (a+b)(a+b) = a^2+2ab+b^2$ algebraic multiplication
3. $(n^2)^2+(2^n)^2+2(n^2)(2^n)-2(n^2)(2^n)$ adding $+2ab-2ab$ to expression
4. $(n^2)^2+2(n^2)(2^n)+(2^n)^2-2(n^2)(2^n)$ re-arrange the expression
5. $(n^2+2^n)^2-2(n^2)(2^n)$ from step 2
6. $(n^2+2^n)^2-2^{n+1}n^2$ law of Exponential
• We try to get the $(n^2+2^n)^2-2^{n+1}n^2$ to conform to $a^2-b^2$ because $a^2-b^2=(a+b)(a-b)$ algebraic multiplication, again
1. Treat $n^2+2^n$ as $a$
2. Since $n$ is odd, n+1 is even. So, we can assume $2m=n+1$, where $m$ is integer
3. So, re-write the $2^{n+1}n^2$ to be $2^{2m}n^2$
4. $2^{2m}n^2=(n2^m)^2$ associative law
5. Treat $n2^m$ as $b$
• It implies that both $a$ and $b$ are both positive integer
• From $a^2-b^2=(a+b)(a-b)$ and the result of $n^4 + 2^4$, it implies that $a$ is greater than $b$
• Hence both $(a+b)$ and $(a-b)$ are positive integer, that causes the result of $n^4 + 2^4$ is combination of $(a+b)$ and $(a-b)$
-
Some interesting factorizations of a polynomial of type $x^4+\text{const}$: $$x^4+4=(x^2+2x+2)(x^2-2x-2) \tag{1}$$
$$x^4+1=(x^2+\sqrt[]{2}x+1)(x^2-\sqrt[]{2}x+1) \tag{2}$$
So one can ask, how to select the coefficients $a,b,c,d$ in
$$(x^2+ax+b)(x^2+cx+d) \tag{3}$$
such that all coefficients of the resulting polynomial are zero except the constant term and the coefficient of the 4th power. The latter is $1$.
If we expand $(3)$ we get
$$x^4+(c+a)x^3+(d+a c+b)x^2+(a d+b c)x+b d$$
And the coefficients disappear, if
$$\begin{eqnarray} c+a &=& 0 \\ d+ ac +b &=& 0 \\ ad+bc &=& 0 \end{eqnarray}$$
When solving for $b,c,d$ we get
$$\begin{eqnarray} c &=& -a \\ b &=& \frac{a^2}{2} \\ d &=& \frac{a^2}{2} \end{eqnarray}$$
and therefore | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9850429116504951,
"lm_q1q2_score": 0.8659972853975705,
"lm_q2_score": 0.879146761176671,
"openwebmath_perplexity": 439.9506829252601,
"openwebmath_score": 0.9091550707817078,
"tags": null,
"url": "http://math.stackexchange.com/questions/853615/proving-an-expression-is-composite"
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.