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help-10
0c14e13593fb4cf4a5e0a2091325234e
356,939,641,485,787,140
BOSS.cYn
11/12/2021 14:11:45
now we are left with 1 (sin^2x - cos^2x)
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Please Correct My Grammar
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help-10
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653,715,253,925,773,300
keepitalphanumeric
11/12/2021 14:11:57
divided by...
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help-10
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BOSS.cYn
11/12/2021 14:12:11
sinx cosx...
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help-10
0c14e13593fb4cf4a5e0a2091325234e
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BOSS.cYn
11/12/2021 14:12:38
would the (sin^2x - cos^2x) be able to be broken up again to be cancelled out?
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help-10
0c14e13593fb4cf4a5e0a2091325234e
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BOSS.cYn
11/12/2021 14:12:55
I think im overthinking it
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help-10
0c14e13593fb4cf4a5e0a2091325234e
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keepitalphanumeric
11/12/2021 14:13:22
note that this is (sin x)(sin x) - (cos x)(cos x)
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help-10
0c14e13593fb4cf4a5e0a2091325234e
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BOSS.cYn
11/12/2021 14:13:35
Ohh so it can be broke up?
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help-10
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BOSS.cYn
11/12/2021 14:14:33
or I guess simplified would be the correct term
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help-10
0c14e13593fb4cf4a5e0a2091325234e
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keepitalphanumeric
11/12/2021 14:15:30
Yes, the cos^2 x is a confusing expression. but it means (cos x) (cos x) not (cos (cos (x)))
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help-10
0c14e13593fb4cf4a5e0a2091325234e
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BOSS.cYn
11/12/2021 14:15:38
gotcha ok
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help-10
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BOSS.cYn
11/12/2021 14:16:04
so then we would be left with sinx - cosx
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help-10
0c14e13593fb4cf4a5e0a2091325234e
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BOSS.cYn
11/12/2021 14:18:43
then for the tanx - cotx we could use the quotient identity to get sinx / cosx - cosx / sin x
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help-10
0c14e13593fb4cf4a5e0a2091325234e
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keepitalphanumeric
11/12/2021 14:20:12
You can go from (sin4 - cos4 )/sin cos directly to sin/cos - cos/sin. (i'm droping the x's because it is easier to write here...)
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help-10
0c14e13593fb4cf4a5e0a2091325234e
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keepitalphanumeric
11/12/2021 14:21:05
Since you will be manipulating equalities all over, the proof will go directly from the first to the last expression.
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help-10
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BOSS.cYn
11/12/2021 14:21:59
oh so then we could keep that sin2 - cos2 because on the right side you need a common denominator and would have to multiply to get sin2 - cos2?
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help-10
0c14e13593fb4cf4a5e0a2091325234e
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keepitalphanumeric
11/12/2021 14:26:03
Yes, but I would argue from the left expression and conclude on the right expression. Left expression = ... (expand the diff of squares) = ... (eliminate the pythagorean) = ... (symplify) = right side (using the definitions).
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help-10
0c14e13593fb4cf4a5e0a2091325234e
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keepitalphanumeric
11/12/2021 14:27:17
This is a clear way to structure the proof that those expressions are equal.
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help-10
0c14e13593fb4cf4a5e0a2091325234e
356,939,641,485,787,140
BOSS.cYn
11/12/2021 14:27:25
oh ok, so how could we break down the right side then?
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help-10
0c14e13593fb4cf4a5e0a2091325234e
356,939,641,485,787,140
BOSS.cYn
11/12/2021 14:27:57
if we've simplifed the left all we can which would be sinx - cosx
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help-10
0c14e13593fb4cf4a5e0a2091325234e
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keepitalphanumeric
11/12/2021 14:29:27
The left expression is (s^2 - c^2) / (s * c) = (s^2)/sc - (c^2)/sc, right?
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help-10
0c14e13593fb4cf4a5e0a2091325234e
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BOSS.cYn
11/12/2021 14:30:29
Yes
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help-10
0c14e13593fb4cf4a5e0a2091325234e
356,939,641,485,787,140
BOSS.cYn
11/12/2021 14:30:51
because you would have to multiply the get the common denominator
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help-10
0c14e13593fb4cf4a5e0a2091325234e
653,715,253,925,773,300
keepitalphanumeric
11/12/2021 14:31:55
Great, then (s^2)/sc - (c^2)/sc = s/c - c/s (which by definition are...)
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help-10
0c14e13593fb4cf4a5e0a2091325234e
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BOSS.cYn
11/12/2021 14:32:58
tan and cot?
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help-10
0c14e13593fb4cf4a5e0a2091325234e
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keepitalphanumeric
11/12/2021 14:33:26
Yes. You have proved that the right side equals the left side.
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help-10
0c14e13593fb4cf4a5e0a2091325234e
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BOSS.cYn
11/12/2021 14:33:44
gotcha, thank you so so much
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help-10
0c14e13593fb4cf4a5e0a2091325234e
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BOSS.cYn
11/12/2021 14:33:57
Walking me through it really helped me
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help-10
0c14e13593fb4cf4a5e0a2091325234e
653,715,253,925,773,300
keepitalphanumeric
11/12/2021 14:34:25
You will get used with this kind of argument, don't worry.
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help-10
0c14e13593fb4cf4a5e0a2091325234e
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BOSS.cYn
11/12/2021 14:34:57
I hope so, just gotta keep working at it!
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help-10
0c14e13593fb4cf4a5e0a2091325234e
356,939,641,485,787,140
BOSS.cYn
11/12/2021 14:35:42
Thank you again!
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help-10
0c14e13593fb4cf4a5e0a2091325234e
356,939,641,485,787,140
BOSS.cYn
11/12/2021 14:35:46
.close
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help-10
7af8f629239b4de7bfbadcc3e4bf4485
519,519,479,546,511,360
Amarinya
04/10/2022 23:10:06
Is $f: \mathbb{R} \to \mathbb{R}$ a bijection if $f(x) = x$? I mean, $f(a) = f(b) \implies a = b$, and for every $b \in \mathbb{R}$, there is an $a \in \mathbb{R}$ (which is $a = b$) so that $f(a) = b$
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help-10
7af8f629239b4de7bfbadcc3e4bf4485
510,789,298,321,096,700
TeXit
04/10/2022 23:10:09
**Taiga**
https://cdn.discordapp.c…479546511381.png
519519479546511381.png
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help-10
7af8f629239b4de7bfbadcc3e4bf4485
456,226,577,798,135,800
Deleted User
04/10/2022 23:14:16
yep
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help-10
7af8f629239b4de7bfbadcc3e4bf4485
519,519,479,546,511,360
Amarinya
04/10/2022 23:15:24
:o
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help-10
7af8f629239b4de7bfbadcc3e4bf4485
519,519,479,546,511,360
Amarinya
04/10/2022 23:16:20
so there's always a bijection $f: X \to X$ whether $X$ is an infinite set or a finite set?
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help-10
7af8f629239b4de7bfbadcc3e4bf4485
510,789,298,321,096,700
TeXit
04/10/2022 23:16:23
**Taiga**
https://cdn.discordapp.c…479546511381.png
519519479546511381.png
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help-10
7af8f629239b4de7bfbadcc3e4bf4485
456,226,577,798,135,800
Deleted User
04/10/2022 23:16:39
yea, the identity function works
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help-10
7af8f629239b4de7bfbadcc3e4bf4485
519,519,479,546,511,360
Amarinya
04/10/2022 23:17:06
that's so coolll
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help-10
7af8f629239b4de7bfbadcc3e4bf4485
519,519,479,546,511,360
Amarinya
04/10/2022 23:17:24
thanks a lot yumeee
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help-10
7af8f629239b4de7bfbadcc3e4bf4485
519,519,479,546,511,360
Amarinya
04/10/2022 23:17:33
.close
1
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help-10
9f52171f96fe4ed8836676b5bc2437c8
456,226,577,798,135,800
Deleted User
04/10/2022 23:17:38
np 🙂
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help-10
9f52171f96fe4ed8836676b5bc2437c8
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Ilyas
04/10/2022 23:22:04
so if I want to prove that 2 sets have the same cardinality, I set up a function, I show that it is one-to-one , and onto. But here is the problem. How do I come up with this function (f(x) = ??) ? Do I have to use my imagination, or is there something I am missing?
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help-10
9f52171f96fe4ed8836676b5bc2437c8
456,226,577,798,135,800
Deleted User
04/10/2022 23:24:47
sometimes coming up with a function for that can be tough 😵‍💫
1
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help-10
9f52171f96fe4ed8836676b5bc2437c8
214,677,699,589,963,780
Ilyas
04/10/2022 23:25:14
yes thats why im confused
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help-10
9f52171f96fe4ed8836676b5bc2437c8
456,226,577,798,135,800
Deleted User
04/10/2022 23:25:17
so yea, I'd say you have to use your imagination sometimes
1
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help-10
9f52171f96fe4ed8836676b5bc2437c8
456,226,577,798,135,800
Deleted User
04/10/2022 23:25:37
there isn't one trick that will always work
1
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help-10
9f52171f96fe4ed8836676b5bc2437c8
214,677,699,589,963,780
Ilyas
04/10/2022 23:25:42
how do I do that on a 1 hour exam lol
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help-10
9f52171f96fe4ed8836676b5bc2437c8
456,226,577,798,135,800
Deleted User
04/10/2022 23:27:10
maybe the problems that could come up on your exam will be somewhat related to other problems you've seen already?
1
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help-10
9f52171f96fe4ed8836676b5bc2437c8
456,226,577,798,135,800
Deleted User
04/10/2022 23:27:57
there aren't any universal tricks but some of the same ideas could work for different problems
1
0
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help-10
9f52171f96fe4ed8836676b5bc2437c8
456,226,577,798,135,800
Deleted User
04/10/2022 23:29:17
what kind of class is this for?
1
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help-10
9f52171f96fe4ed8836676b5bc2437c8
214,677,699,589,963,780
Ilyas
04/10/2022 23:32:29
discrete math
0
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help-10
9f52171f96fe4ed8836676b5bc2437c8
456,226,577,798,135,800
Deleted User
04/10/2022 23:34:32
any examples in particular you are struggling on?
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help-10
9f52171f96fe4ed8836676b5bc2437c8
214,677,699,589,963,780
Ilyas
04/10/2022 23:35:47
https://imgur.com/gallery/gX2TKYh
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help-10
9f52171f96fe4ed8836676b5bc2437c8
456,226,577,798,135,800
Deleted User
04/10/2022 23:36:35
ok, cool
1
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help-10
9f52171f96fe4ed8836676b5bc2437c8
456,226,577,798,135,800
Deleted User
04/10/2022 23:37:08
an idea you could have for this is to make a linear function that sends a to 0, and b to 1
1
0
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help-10
9f52171f96fe4ed8836676b5bc2437c8
214,677,699,589,963,780
Ilyas
04/10/2022 23:37:49
what?
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help-10
9f52171f96fe4ed8836676b5bc2437c8
456,226,577,798,135,800
Deleted User
04/10/2022 23:39:36
one thing that might sounds a little off about what I said is that 0 and 1 are not in S, and a and b are not in W
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help-10
9f52171f96fe4ed8836676b5bc2437c8
456,226,577,798,135,800
Deleted User
04/10/2022 23:39:49
is that confusing or is it something else?
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help-10
9f52171f96fe4ed8836676b5bc2437c8
456,226,577,798,135,800
Deleted User
04/10/2022 23:42:21
but the idea is that W and S are both like lines
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help-10
9f52171f96fe4ed8836676b5bc2437c8
456,226,577,798,135,800
Deleted User
04/10/2022 23:42:31
if you think of them on the real number line
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help-10
9f52171f96fe4ed8836676b5bc2437c8
456,226,577,798,135,800
Deleted User
04/10/2022 23:42:53
and you can transform one into the other by sending the endpoints of one to the endpoints of the other
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help-10
9f52171f96fe4ed8836676b5bc2437c8
456,226,577,798,135,800
Deleted User
04/10/2022 23:44:00
and transforming all the other stuff in each of the sets
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help-10
9f52171f96fe4ed8836676b5bc2437c8
456,226,577,798,135,800
Deleted User
04/10/2022 23:44:11
but that part kinda happens naturally when you do this
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help-10
9f52171f96fe4ed8836676b5bc2437c8
214,677,699,589,963,780
Ilyas
04/10/2022 23:45:12
so f: W -> S and f: S -> W are the same?
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help-10
9f52171f96fe4ed8836676b5bc2437c8
456,226,577,798,135,800
Deleted User
04/10/2022 23:45:48
hmm I'm not sure exactly what you are asking?
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help-10
9f52171f96fe4ed8836676b5bc2437c8
214,677,699,589,963,780
Ilyas
04/10/2022 23:46:11
like the definition of the function
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help-10
9f52171f96fe4ed8836676b5bc2437c8
214,677,699,589,963,780
Ilyas
04/10/2022 23:46:19
that I will use in this problem
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help-10
9f52171f96fe4ed8836676b5bc2437c8
456,226,577,798,135,800
Deleted User
04/10/2022 23:47:40
uhhh
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help-10
9f52171f96fe4ed8836676b5bc2437c8
456,226,577,798,135,800
Deleted User
04/10/2022 23:48:00
maybe I'll say... no, but I'm still not sure what you mean 😕
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help-10
9f52171f96fe4ed8836676b5bc2437c8
214,677,699,589,963,780
Ilyas
04/10/2022 23:48:33
how would you solve this?
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help-10
9f52171f96fe4ed8836676b5bc2437c8
214,677,699,589,963,780
Ilyas
04/10/2022 23:48:38
like your method
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help-10
9f52171f96fe4ed8836676b5bc2437c8
456,226,577,798,135,800
Deleted User
04/10/2022 23:49:54
this is how I would come up with the function
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help-10
9f52171f96fe4ed8836676b5bc2437c8
214,677,699,589,963,780
Ilyas
04/10/2022 23:53:07
yes but can you write the function youre thinking of?
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help-10
9f52171f96fe4ed8836676b5bc2437c8
456,226,577,798,135,800
Deleted User
04/10/2022 23:53:13
sure
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help-10
9f52171f96fe4ed8836676b5bc2437c8
456,226,577,798,135,800
Deleted User
04/10/2022 23:54:41
I think it's going to be $f:(0,1)\to (a,b)$ defined by $f(x)=(b-a)x+a$
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help-10
9f52171f96fe4ed8836676b5bc2437c8
510,789,298,321,096,700
TeXit
04/10/2022 23:54:43
**yume ♡**
https://cdn.discordapp.c…931315097620.png
465430931315097620.png
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9f52171f96fe4ed8836676b5bc2437c8
214,677,699,589,963,780
Ilyas
04/10/2022 23:55:20
ok so what made you put (b-a) for m and a for c ?
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help-10
9f52171f96fe4ed8836676b5bc2437c8
456,226,577,798,135,800
Deleted User
04/10/2022 23:55:42
it's like a high school algebra problem, finding a line that passes through two points
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help-10
9f52171f96fe4ed8836676b5bc2437c8
456,226,577,798,135,800
Deleted User
04/10/2022 23:55:51
the two points are (0,a) and (1,b)
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help-10
9f52171f96fe4ed8836676b5bc2437c8
214,677,699,589,963,780
Ilyas
04/10/2022 23:56:04
I totaly forgot how to do that lol
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help-10
9f52171f96fe4ed8836676b5bc2437c8
456,226,577,798,135,800
Deleted User
04/10/2022 23:56:23
haha, happens
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help-10
9f52171f96fe4ed8836676b5bc2437c8
214,677,699,589,963,780
Ilyas
04/10/2022 23:57:13
whats the formula again?
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help-10
9f52171f96fe4ed8836676b5bc2437c8
214,677,699,589,963,780
Ilyas
04/10/2022 23:57:31
is it like m = x1 -x2/y1 - y2
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help-10
9f52171f96fe4ed8836676b5bc2437c8
456,226,577,798,135,800
Deleted User
04/10/2022 23:58:06
uhh you can write f(x) = mx + c and replace f(x) with a and x with 0
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help-10
9f52171f96fe4ed8836676b5bc2437c8
456,226,577,798,135,800
Deleted User
04/10/2022 23:58:26
then a = c
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help-10
9f52171f96fe4ed8836676b5bc2437c8
456,226,577,798,135,800
Deleted User
04/10/2022 23:58:42
then do the same with (1,b)
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help-10
9f52171f96fe4ed8836676b5bc2437c8
214,677,699,589,963,780
Ilyas
04/10/2022 23:58:51
oh
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help-10
9f52171f96fe4ed8836676b5bc2437c8
214,677,699,589,963,780
Ilyas
04/10/2022 23:58:55
I see thanks
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help-10
9f52171f96fe4ed8836676b5bc2437c8
456,226,577,798,135,800
Deleted User
04/11/2022 00:00:04
yea you could do that too but I remember my way easier 😛
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help-10
9f52171f96fe4ed8836676b5bc2437c8
214,677,699,589,963,780
Ilyas
04/11/2022 00:03:38
ok one last question, why exactly send a to 0, and b to 1? why didnt you say b to 0 and a to 1
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help-10
9f52171f96fe4ed8836676b5bc2437c8
456,226,577,798,135,800
Deleted User
04/11/2022 00:03:50
you could do that too
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help-10
9f52171f96fe4ed8836676b5bc2437c8
214,677,699,589,963,780
Ilyas
04/11/2022 00:04:11
okay
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help-10
9f52171f96fe4ed8836676b5bc2437c8
214,677,699,589,963,780
Ilyas
04/11/2022 00:04:12
thanks
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help-10
9f52171f96fe4ed8836676b5bc2437c8
456,226,577,798,135,800
Deleted User
04/11/2022 00:16:09
@Ilyas are you ok with all of this? do you have any more questions?
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help-10
6207a49b94364d2283c73a7d27358a12
659,663,332,521,017,300
EroxMg
04/11/2022 00:49:02
Hey
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6207a49b94364d2283c73a7d27358a12
800,622,149,903,056,900
Shen
04/11/2022 00:49:20
:JustAsk:
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help-10
6207a49b94364d2283c73a7d27358a12
659,663,332,521,017,300
EroxMg
04/11/2022 00:49:26
https://cdn.discordapp.c…611/IMG_3256.jpg
IMG_3256.jpg
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help-10
6207a49b94364d2283c73a7d27358a12
659,663,332,521,017,300
EroxMg
04/11/2022 00:49:33
This is Quadratic Equations
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help-10
6207a49b94364d2283c73a7d27358a12
659,663,332,521,017,300
EroxMg
04/11/2022 00:49:47
I just need to understand what it means by “factored over true integers”
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