instruction stringlengths 12 30k |
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in trying to find this limit, all the (x,y) paths i check approaching (0,0) produce the same result, namely, the function approaches 0. consequently, i came up with the simplest proof i could that the limit indeed exists and is zero. however [wolframalpha tells me][1] that the limit doesnt exist. assuming im wrong, can... |
Suppose $f$ is an unknown polynomial of degree $n$ (in one indeterminate) but the sequence $\{ f(k) \}_{k \in \mathbb{N}}$ is given. It is a nice exercise to show that one needs only the first $n+1$ terms of the sequence to determine the coefficients of $f$. That is, simply solve the matrix equation $A\mathbf{x} = b$, ... |
A graph can be thought of as a pair $(V,E)$, where $V$ are the vertices and $E$ are the edges. An **isomorphism** between two graphs, $G_1 = (V_1,E_1)$ and $G_2=(V_2,E_2)$ means a function $f\colon V_1\to V_2$ that is bijective between the vertices (one-to-one and onto, so every vertex in $G_1$ is mapped to a vertex of... |
Is it algebraic the number 0,2468101214 ...? (After the comma, the natural numbers are juxtaposed pairs).
|
0,246810121416...: Is it a algebraic number? |
A graph can be thought of as a pair $(V,E)$, where $V$ are the vertices and $E$ are the edges. An **isomorphism** between two graphs, $G_1 = (V_1,E_1)$ and $G_2=(V_2,E_2)$ means a function $f\colon V_1\to V_2$ that is bijective between the vertices (one-to-one and onto, so every vertex in $G_1$ is mapped to a vertex of... |
Is it algebraic the number 0.2468101214 ...? (After point, the natural numbers are juxtaposed pairs).
|
0.246810121416...: Is it a algebraic number? |
I'll give some hints:
1) Notice that $X_i^2 = X_i$, since $X_i$ is $0$ or $1$. Then,
$E X_i = 0 \cdot \mathrm{P}(X_i=0) + 1 \cdot \mathrm{P}(X_i = 1) = \mathrm{P}(X_i = 1)$. What is this probability?
2) Again, $X_i X_j$ is $0$ or $1$. Notice that $X_i X_j = 1$ if and only if person $i$ gets their hat AND pe... |
Let X be an infinite-dimensional Banach space, and T a compact operator from X to X. Why must 0 then be a spectral value for T?
I believe this is equivalent to saying that T is not bijective, but I am not sure how to show that injectivity implies the absence of surjectivity and the other way around (or if this is e... |
Please note:
- a) $X_{i}=1$ probability $\frac{1}{n}$ and 0, otherwise. Thus $X_{i}^{2}=1$ with probability $\frac{1}{n}$ and 0, otherwise, which *implies* $E(X_{i}^{2})=\frac{1}{n}$
- b) The probability that both $X_{i}$ and $X_{j}$ are 1 is $\frac{1}{n(n-1)}$ (Why?)
- c) $E(S_{n}^{2}) = \sum\limits_{i} E[X_{... |
Does it make sense to say that there are more irrational numbers than rational? |
Are there many more irrational numbers than rational? |
Hey guys, second post for today. I have a midterm on tuesday so I am just practicing some proofs from the textbook.
The question states:
If $a \in ZZ$, prove that $a^2$ is not congruent to 2 modulo 4 or 3 modulo 4.
Our work:
Its an if then proof, so I am thinking we can do a contradiction proof.
Suppose $a... |
Suppose $f$ is an unknown polynomial of degree $n$ (in one indeterminate) but the sequence $\{ f(k) \}_{k \in \mathbb{N}}$ is given. It is a nice exercise to show that one needs only the first $n+1$ terms of the sequence to determine the coefficients of $f$. That is, simply solve the matrix equation $A\mathbf{x} = b$, ... |
I figured out this order on omega^2: elements of different columns are ordered by their column number; within column m, the order is 0 > 1 > 2 > ... > n, where n is the first such that P(m,n) is true; then 0 < n+1 < n+2 < ... The column is an infinite descending sequence if there is no such n. This order is well-founde... |
The title is somewhat deceptive: I know that $|\mathbb{Q}|=|\mathbb{Z}|.$
But suppose I wanted to compare the sets, knowing that they are of the same cardinality but still wondering if there was some sense in which Q could be larger. There are two usual ways of thinking about this:
$\mathbb{Z}\subset\mathbb{Q}$ ... |
Are there many more rational numbers than integers? |
Hey guys, second post for today. I have a midterm on tuesday so I am just practicing some proofs from the textbook.
The question states:
If $a \in \mathbb{Z}$, prove that $a^2$ is not congruent to 2 modulo 4 or 3 modulo 4.
Our work:
It's an if then proof, so I am thinking we can do a contradiction proof.
S... |
Suppose $f$ is an unknown polynomial of degree $n$ (in one indeterminate) but the sequence $\{ f(k) \}_{k \in \mathbb{N}}$ is given. It is a nice exercise to show that one needs only the first $n+1$ terms of the sequence to determine the coefficients of $f$. That is, simply solve the matrix equation $A\mathbf{x} = b$, ... |
If $f\colon[0,1]\to\mathbb{R}$ is cadlag ([continu à droite, limites à gauche][1]) then we can ask what the possible jumps are. That is, for what functions $g\colon(0,1]\to\mathbb{R}$ is there a cadlag function $f$ with $g(x)=\Delta f(x)\equiv f(x)-f(x-)$?
The answer is that $g$ occurs as the jumps of ... |
Suppose $f$ is an unknown polynomial of degree $n$ (in one indeterminate) but the sequence $\{ f(k) \}_{k \in \mathbb{N}}$ is given. It is a nice exercise to show that one needs only the first $n+1$ terms of the sequence to determine the coefficients of $f$. That is, simply solve the matrix equation $A\mathbf{x} = b$, ... |
Suppose $f$ is an unknown polynomial of degree $n$ (in one indeterminate) but the sequence $\{ f(k) \}_{k \in \mathbb{N}}$ is given. It is a nice exercise to show that one needs only the first $n+1$ terms of the sequence to determine the coefficients of $f$. That is, simply solve the matrix equation $A\mathbf{x} = b$, ... |
I figured out this order on $\omega^2$: elements of different columns are ordered by their column number; within column $m$, the order is $0 \gt 1 \gt 2 \gt \cdots \gt n$, where $n$ is the first such that $P(m,n)$ is true; then $0 \lt n+1 \lt n+2 \lt \cdots$. The column is an infinite descending sequence if there is no... |
Consider a $G$-equivariant map $\pi:X\to Y$ for $G$ an affine algebraic group, such that $\pi$ is a good categorical quotient. Is there any relationship between $H^*_G(X)$ and $H^*(Y)$? Is there if $\pi$ is a good geometric quotient, or if the quotient space is smooth? |
Suppose $f$ is an unknown polynomial of degree $n$ (in one indeterminate) but the sequence $\{ f(k) \}_{k \in \mathbb{N}}$ is given. It is a nice exercise to show that one needs only the first $n+1$ terms of the sequence to determine the coefficients of $f$. That is, simply solve the matrix equation $A\mathbf{x} = b$, ... |
Combining the answers given by me and Ralth to the probability question at http://math.stackexchange.com/questions/10284/probability-question, we get the following identity:
$$
\sum\limits_{k = m}^n {{n \choose k}p^k (1 - p)^{n - k} {k \choose m} p_j^m (1 - p_j )^{k - m} } = {n \choose m} (p p_j)^m (1 - p p_j)^{n-m},... |
Probability question leading to an interesting identity? |
Suppose $f$ is an unknown polynomial of degree $n$ (in one indeterminate) but the sequence $\{ f(k) \}_{k \in \mathbb{N}}$ is given. It is a nice exercise to show that one needs only the first $n+1$ terms of the sequence to determine the coefficients of $f$. That is, simply solve the matrix equation $A\mathbf{x} = b$, ... |
I have a finite alphabet A and a set S of $n$ ordered pairs $(a, b)$ with $a,b\in A.$ I want to find out how many sequences $(a_1, b_1),\ldots,(a_n,b_n)$ there are where adjacent elements have either $a_{k-1}=a_k$ or $b_{k-1}=b_k.$
Any ideas on how to go about finding this? Formulas, generating functions, algorith... |
I have a finite alphabet A and a sequence S of $n$ ordered pairs $(a, b)$ with $a,b\in A.$ I want to find out how many sequences $(a_1, b_1),\ldots,(a_n,b_n)$ there are where adjacent elements have either $a_{k-1}=a_k$ or $b_{k-1}=b_k.$
Any ideas on how to go about finding this? Formulas, generating functions, alg... |
Two vectors **x**=(x1,x2) and **y**=(y1,y2) are orthogonal if their scalar product is zero, i.e. x1*y1+x2*y2=0. What surface does this equation describe in the four-dimensional (x1,x2,y1,y2) space? How about the orthogonality condition for 3-dimensional (or higher dimensional) vectors? What is the corresponding surface... |
Combining the answers given by me and Ralth to the probability question at http://math.stackexchange.com/questions/10284/probability-question, we get the following identity:
$$
\sum\limits_{k = m}^n {{n \choose k}p^k (1 - p)^{n - k} {k \choose m} p_j^m (1 - p_j )^{k - m} } = {n \choose m} (p p_j)^m (1 - p p_j)^{n-m},... |
I figured out this order on $\omega^2$: elements of different columns are ordered by their column number; within column $m$, the order is $0 \gt 1 \gt 2 \gt \cdots \gt n$, where $n$ is the first such that $P(m,n)$ is true; then $0 \lt n+1 \lt n+2 \lt \cdots$. The column is an infinite descending sequence if there is no... |
Two vectors $\mathbf{x}=(x_1,x_2)$ and $\mathbf{y}=(y_1,y_2)$ are orthogonal if their scalar product is zero, i.e. $x_1 y_1 + x_2 y_2=0$. What surface does this equation describe in the four-dimensional $(x_1,x_2,y_1,y_2)$ space? How about the orthogonality condition for $3$-dimensional (or higher dimensional) vectors?... |
How would I go about creating a polynomial $f$ such that
\begin{align*}
f(x) &= A_1\\\
f(y) &= A_2\\\
f'(x) &= B_1\\\
f'(y) &= B_2
\end{align*}
With $x$ not equal to $y$ and $A_1$, $A_2$, $B_1$, $B_2$ real numbers?
|
There are two equivalent definitions of a locally convex topological vector space. Note that since the vector space is topological, addition by a fixed vector x and multiplication by a fixed scalar r are both homeomophisms, which means it is enough to give a neighborhood basis at the vector 0.
The first definition i... |
What is an example of a topological vector space which contains a non-absorbent or non-balanced open convex set around the origin? |
I'm going through an examples section (on improper integrals) but I got lost at this bit:
lim t->-infinity (t/e^-t) = lim t->-infinity (1/-e^-t)
I think it's a simple algebra trick but I don't see how the right hand side came to be. How did it become 1/-e^-t? |
In general, there are many possible answers. But, suppose your polynomial looks like this:
$$f(X) = a_n X^n + \cdots + a_1 X + a_0$$
with $a_0,\ldots,a_n$ real numbers. Then
$$f'(X) = na_n X^n + \cdots + a_1.$$
Evaluating at $x$ and at $y$ will give you a bunch of equations that they need to satisfy:
\begin{align... |
There are two equivalent definitions of a locally convex topological vector space. Note that since the vector space is topological, addition by a fixed vector x and multiplication by a fixed scalar r are both homeomophisms, which means it is enough to give a neighborhood basis at the vector 0.
The first definition i... |
In general, there are many possible answers. But, suppose your polynomial looks like this:
$$f(X) = a_n X^n + \cdots + a_1 X + a_0$$
with $a_0,\ldots,a_n$ real numbers. Then
$$f'(X) = na_n X^n + \cdots + a_1.$$
Evaluating at $x$ and at $y$ will give you a bunch of equations that they need to satisfy:
\begin{align... |
I'm going through an examples section (on improper integrals) but I got lost at this bit:
$$\lim_{t\to-\infty} \frac{t}{e^{-t}} = \lim_{t\to-\infty}\frac{1}{-e^{-t}}.$$
I think it's a simple algebra trick but I don't see how the right hand side came to be. How did it become $1/-e^{-t}$? |
In general, there are many possible answers. But, suppose your polynomial looks like this:
$$f(X) = a_n X^n + \cdots + a_1 X + a_0$$
with $a_0,\ldots,a_n$ real numbers. Then
$$f'(X) = na_n X^n + \cdots + a_1.$$
Evaluating at $x$ and at $y$ will give you a bunch of equations that they need to satisfy:
\begin{align... |
In general, there are many possible answers. But, suppose your polynomial looks like this:
$$f(X) = a_n X^n + \cdots + a_1 X + a_0$$
with $a_0,\ldots,a_n$ real numbers. Then
$$f'(X) = na_n X^n + \cdots + a_1.$$
Evaluating at $x$ and at $y$ will give you a bunch of equations that they need to satisfy:
\begin{align... |
In general, there are many possible answers. But, suppose your polynomial looks like this:
$$f(X) = a_n X^n + \cdots + a_1 X + a_0$$
with $a_0,\ldots,a_n$ real numbers. Then
$$f'(X) = na_n X^n + \cdots + a_1.$$
Evaluating at $x$ and at $y$ will give you a bunch of equations that they need to satisfy:
\begin{align... |
In general, there are many possible answers. But, suppose your polynomial looks like this:
$$f(X) = a_n X^n + \cdots + a_1 X + a_0$$
with $a_0,\ldots,a_n$ real numbers. Then
$$f'(X) = na_n X^n + \cdots + a_1.$$
Evaluating at $x$ and at $y$ will give you a bunch of equations that they need to satisfy:
\begin{align... |
In general, there are many possible answers. But, suppose your polynomial looks like this:
$$f(X) = a_n X^n + \cdots + a_1 X + a_0$$
with $a_0,\ldots,a_n$ real numbers. Then
$$f'(X) = na_n X^n + \cdots + a_1.$$
Evaluating at $x$ and at $y$ will give you a bunch of equations that they need to satisfy:
\begin{align... |
<html>
<body>
Informal justification of positive x negative = negative:<br>
<br>
Continue the pattern:<br>
2x3=6<br>
2x2=4<br>
2x1=2<br>
2x0=0<br>
2x-1= ? (Ans=-2)<br>
2x-2= ? (Ans=-4)<br>
2x-3= ? (Ans=-6)<br>
<br>
The number on the right-hand side keeps decreasing by 2.<br>
<br>
Informal justification o... |
How to describe all ring homomorphism from Z*Z to Z ?
|
Specifically, if you have a topos with natural numbers object (assume choice if you want to), what familiar theorems don't hold? I've heard that showing the Dedekind reals are the same as the Cauchy reals is one. Where in the arguments is well-pointedness used? It seems hard to find examples of this. |
Where is the well-pointedness assumption of ETCS used in everyday math? |
How to describe all ring homomorphism from $Z \times Z \to Z$ ?
|
For a sequence defined by a formula normally the usual limit rules allows
one to find its limit. But for a sequence defined by a recurrence, up to now
I have only seen some refined ad-hoc methods, mostly in Problems.
The Trick explains that "[There is one trick that is (...) first to prove
that a limit exists, an... |
I'm in a little bit of a pickle here, can someone tell me if it is actually possible to solve these? From the general rule of thumb, 4 variables, 4 equations should be able to right? |
Is it possible to solve: xa = 4200, xb = 4410, ya=4000, yb = 4200 for x, y, a and b? |
It is known that, in the universe of complex numbers, the only root of the equation x² - 2x 1 = 0 is 1. Could we say that the equation has two equal real roots? Or should we say that the equation has one real root with multiplicity 2?
|
A root? Or two roots? |
It is known that, in the universe of complex, the only root of the equation
x² - 2x + 1 = 0 is 1. Could we say that the equation has two equal real roots? Or should we say that the equation has one real root with multiplicity 2?
|
It is known that, in the universe of complex numbers, the only root of the equation x² - 2x + 1 = 0 is 1. Could we say that the equation has two equal real roots? Or should we say that the equation has one real root with multiplicity 2?
|
For a sequence defined by a formula normally the usual limit rules allows
one to find its limit. But for a sequence defined by a recurrence, up to now
I have only seen some refined ad hoc methods, mostly in Problems.
The Trick explains that "[There is one trick that is (...) first to prove
that a limit exists, an... |
I'm stuck with this problem I'm trying to solve from about an hour. Here's the question.
What is the remainder when (3^202)+137 is divided by 101?
There are 4 options -> 36, 45, 56, 11
I want to know the answer of the question with proper and possibly easiest method to solve the problem.
Thanks in advance, ... |
What will be the remainder? |
This is only a first approximation to your answer, but I think it's a good start.
Let $x$ and $y$ be vectors in $\mathbb{R}^n$ and assume $x\cdot y = 0$. Now, I'm going to add an additional condition that neither $x$ nor $y$ is $0$. To give away the punchline, it will turn out there's a nice description of <i>thes... |
I'm stuck with the dual of an exercise in Atiyah-MacDonald.
It's easily seen that tensoring over $R$ with $-\otimes M$ the sequence
$$
0\to \mathfrak a\to R\to R/\mathfrak a\to 0
$$
one gets the isomorphism $R/\mathfrak a\otimes M\cong M/\mathfrak aM$.
What can I deduce from "*Hom*(-,M)ing" the same sequence? I... |
**I think Chandru1 is correct. The answer is 45.**
How did I get it?
1) The remainder of 3^30 / 101 = 6
2) 6^6 = 46656.
3) 46656 * (The remainder of 3^22 / 101) = 2286144.
4) 2286144 + 137 = 2286281
5) The remainder of 2286281 / 101 = 45
**Note in step 2 and 3 that 30 * 6 = 180. And 180... |
i have to solve the following recurrence :T(n)=2*T(n/2)+T(n/3)+θ(n^2) i have done the whole tree analyses and now i have to prove that T(n)<=d*n^2*log2(n) but i can not find mathematical way to prove that ,is there any trick with logarithms that i can use? |
Mathematicians and physicists use very different languages when they talk about tensors. Fortunately, they are talking about the same thing, but unfortunately, this is not obvious at all. Let me explain.
For simplicity, I'm going to focus on **covariant 2-tensors**, since this case already contains the main intu... |
Let's say I have a truth table like this:
X Y A
t t f
t f t
f t t
f f f
Now, I have to find the formula for A. This case is rather easy because I can immediately see that this looks like the inverted values of X~B, thus ¬(X~B). Now, for a more complicated problem I do not know... |
How to find the logical formula for a given truth table? |
Asking «what is $M/\operatorname{im}\alpha^*$?» is *not* the dual of the A-M exercise.
If you really want a dual statement, you should be looking for a description of $\hom(R/\mathfrak a,M)$. |
Let's say I have a truth table like this:
X Y A
t t f
t f t
f t t
f f f
Now, I have to find the formula for A. This case is rather easy because I can immediately see that this looks like the inverted values of X~B, thus ¬(X~B). Now, for a more complicated problem I do not know... |
I believe I've found all Amicable Pairs (n, m) such that n is in the closed interval [1, 34000000]. Here's my [list][1].
By inspection I believe that for all n in the list, n is odd iff its last digit is 5. Can it be proved that this is true for all Amicable Pairs?
[1]: http://tutoree7.pastebin.com/fg3... |
True?: Let (n, m) be an arbitrary Amicable Pair. Then n is odd iff its last digit is 5 |
I have to solve the following recurrence :$\displaystyle T(n)=2T(n/2)+T(n/3)+\theta(n^2)$
I have done the whole tree analyses and now I have to prove that $\displaystyle T(n) \leq dn^{2}\log_{2}(n)$ but I can not find mathematical way to prove that, is there any trick with logarithms that I can use? |
Let $R$ be a commutative ring and $I_1, \cdots, I_n$ pairwise comaximal ideals in $R$, e.g. $I_i + I_j = R$ for $i \neq j$.
Why are the ideals $I_1^{n_1}, ... , I_r^{n_r}$ (for any $n_1,...,n_r \in\mathbb N$) also comaximal? |
How about 87633 and 69615? Found 87633 in [OEIS A063990][1]
[1]: http://oeis.org/A063990 |
Consider a $G$-equivariant map $\pi:X\to Y$ for $G$ an affine algebraic group, such that $\pi$ is a good categorical quotient. Is there any relationship between $H^*_G(X)$ and $H^*(Y)$? Is there if $\pi$ is a good geometric quotient, or if the quotient space is smooth?
EDIT: A categorical quotient is an equivarian... |
Given a $1$ dimensional domain $R$ with fraction field $K$. Suppose $\mathfrak{m}$ is a maximal ideal of $R$. Then, I need to show that $R_m$ is a maximal subring of $K$ in the sense that if $x\in K-R_\mathfrak{m}$, then, $R_m[x]=K$.
1)Is this true? How would you go about showing this?
2)If this is true, can any ... |
Given a $1$ dimensional domain $R$ with fraction field $K$. Suppose $\mathfrak{m}$ is a maximal ideal of $R$. Then, I need to show that $R_m$ is a maximal subring of $K$ in the sense that if $x\in K-R_\mathfrak{m}$, then, $R_m[x]=K$.
1)Is this true? How would you go about showing this?
2)If this is true, can any ... |
Imagine we have a wizard that knows a few spells. Each spell has 3 attributes:
Damage, cooldown time, and a cast time.
Cooldown time: the amount of time (t) it takes before being able to cast that spell again. A spell goes on "cooldown" the moment it begins casting.
Cast time: the amount of time (t) it takes to ... |
Imagine we have a wizard that knows a few spells. Each spell has 3 attributes:
Damage, cooldown time, and a cast time.
Cooldown time: the amount of time (t) it takes before being able to cast that spell again. A spell goes on "cooldown" the moment it begins casting.
Cast time: the amount of time (t) it takes to ... |
If you have a function, implicitly defined, how would understand the following;
Assume y=y(x)=, find y'(0) and y''(0).
After I implicitly derivate it the first time, I have an expression with dy/dx=myDerivative. Do I just fill inn zero for x, remove the parts that gets canceled out? I'm not sure if the answer i... |
Easy; understanding question about implicit diff? |
I have a finite alphabet A and a set S of $n$ ordered pairs $(a, b)$ with $a,b\in A.$ I want to find out how many sequences $(a_1, b_1),\ldots,(a_n,b_n)$ there are where adjacent elements have either $a_{k-1}=a_k$ or $b_{k-1}=b_k.$ (Since S is a set, they can't have both.)
Any ideas on how to go about finding this?... |
Given a finite abelian group G. H,K subgroups of G such that G/H and G/K are primary cyclic. H and K intersect trivially (H/\K=1). H!=1 and K!=1.
Is it then true that HK=G?
My attempt so far: It's easy to show that [G:HK] | ([G:H], [G:K]). So if G/H and G/K are primary cyclic for different primes, then we're done... |
G finite abelian. G/H, G/K primary cyclic. H/\K=1. H,K!=1. Is HK=G? |
There are lot of resources on the web and i think you can figure out your problem by reading some sources. Here i give some files in which the concept of *implicit differentiation* has been written nicely.
- http://www.mash.dept.shef.ac.uk/Resources/web-implicit.pdf
- http://www.youtube.com/watch?v=ximF06lmqPM
... |
Given a finite abelian group $G$. $H$ and $K$ subgroups of $G$ such that $G/H$ and $G/K$ are primary cyclic. $H$ and $K$ intersect trivially ($H\cap K=1$), and $H\neq 1$, $K\neq 1$.
Is it then true that $HK=G$?
My attempt so far: It's easy to show that $[G:HK] | ([G:H], [G:K])$. So if $G/H$ and $G/K$ are primar... |
$G$ finite abelian. $G/H$, $G/K$ primary cyclic. $H\cap K=1$. $H\neq 1\neq K$. Is $HK=G$? |
If you have a function, implicitly defined, how would understand the following;
Assume $y=y(x)$, find $y'(0)$ and $y''(0)$.
After I implicitly derivate it the first time, I have an expression with $dy/dx$. Do I just fill in zero for $x$, remove the parts that gets canceled out? I'm not sure if the answer is a n... |
I have a finite alphabet A and a set S of $n$ ordered pairs $(a, b)$ with $a,b\in A.$ I want to find out how many sequences $(a_1, b_1),\ldots,(a_n,b_n)$ there are where adjacent elements have either $a_{k-1}=a_k$ or $b_{k-1}=b_k.$ (Since S is a set, they can't have both.)
Any ideas on how to go about finding this?... |
What is x if 2x-sin2x=pi/2, 0<x<pi/2? |
What is x if 2x-sin2x=pi/2, x in the first quadrant? |
All power sets contain an empty set, so does this make the power set itself an empty set? Or am I mistaken? |
Is a power set null because it contains an empty set? |
Rubin [1] gives an algorithm for counting Hamiltonian paths.
[1] Frank Rubin, <a href="http://portal.acm.org/citation.cfm?id=321854">A search procedure for Hamilton paths and circuits</a> (1974) |
Let a statement P is "X is true if and only if Y is true". What is the negation of P? I am little confused. It seems that digital equivalent of this statement is P = X and Y. Hence negation of P is (not X) or (not Y) i.e. Either X or Y is false. Am I right guys? |
Negation of if and only if? |
What is $x$ if $2x-\sin2x=\pi/2$, $x$ in the first quadrant? |
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