Question

proof problem

Answer 1

Suppose that \[f: X \rightarrow Y\] and that \[x \in X\]

Answer 2

Justify the statement \[x \in f^{-1}({f(x)})\]
Is \[{x}= \bar f^{-1} ({f(x)})\] true?

Answer 3

\[\bar f^{-1}: \mathcal P \left({Y}\right) \rightarrow \mathcal P \left({X}\right)\] is defined by \[\bar f^{-1} (B) = [x \in X: f(x) \in B] \]

Answer 4

https://www.writelatex.com/746297qjgdvp#/1618778/ for the whole thing... go to texmaker and see ^^

Answer 5

@wio

Answer 6

Justify the statement: $x \in f^{-1}({f(x)}).$ \\

$X$ belongs in the inverse of $f(x)$. The $f(x)$ is the image of elements in the domain.
By Definition 5.1.8, if we let $f : X \rightarrow Y$ then the inverse of $f$ (or $f$ inverse denoted $f^{-1})$ is the pairing defined by the rule that if $f(x)=y$, then $f^{-1}(y)=x$.[...] all images of elements of the domain are in ${f(x) : x\in X}$

Answer 7

ughhhhh which latex would work and not piece by piece

Answer 8

change $ into `\(` and `\)`

Answer 9

So your question is whether or not x = f^-1(f(x)) is true?

Answer 10

For the \(=\) one, you just have to let \(x\) be something other than a set, since the inverse seems to return sets.

Answer 11

Justify the statement: \( x \in f^{-1}({f(x)}). \)


X$ belongs in the inverse of \( f(x) \) The \( f(x) \) is the image of elements in the domain. By Definition 5.1.8, if we let \( f : X \rightarrow Y \) then the inverse of \( f (or \( f \) inverse denoted \( f^{-1}) \) is the pairing defined by the rule that if \( f(x)=y \), then \( f^{-1}(y)=x \).[...] all images of elements of the domain are in \( {f(x) : x\in X} \)

Answer 12

ugh I wish os adopted the $ sign for latex.

Answer 13

so for \[\bar f^{-1} (B) = [x \in X: f(x) \in B]\]

let's have B = -5 and X = 5

Answer 14

then . . . get the inverse?

Answer 15

They define f(B) = abs(B) and f(x) = abs(x).

Answer 16

right

Answer 17

but do I just plug it in or no
?

Answer 18

Well, what i'm wondering is why consider the abs(x)? Is the abs(x) bijective?

Answer 19

bijective?

Answer 20

wait a sec... my class didn't go over that

Answer 21

Right. That is, does the abs(x) have an inverse? A bijective function is one which is both injective and surjective.

Answer 22

I'm given this \[{x}= \bar f^{-1} ({f(x)})\] and I need to prove that's it's true or false. ... so the function has an inverse and a regular function right?

Answer 23

Yes, if f^-1 denotes the inverse of f, then f is bijective.

Answer 24

so this is asking whether or not MY X is equal to the INVERSE of a FUNCTION

Answer 25

hmm if we let f(x ) = 5 then I would have f^-1 (5)

Answer 26

but where does the f^-1 (5) go?

Answer 27

Let's take an example like f(x) = x. Then f^-1(x) = x. Let x = 5 and observe that f(5) = 5 and f^-1(5) = 5. This implies that f^-1(f(5)) = f^-1(5) = 5

Answer 28

damn it so the statement is true.

Answer 29

I think the question should say \[
\{x\}= f^{-1}(f(x))
\]

Answer 30

Their definition of the inverse returns a set that is a subset of \(X\). It makes no sense to compare \(x\) which likely is not a set to begin with.

Answer 31

We can make a diagram like this too, Isn't the question whether or not it is true that f^-1(f(x)) = x ? The notation for f(B) and f(X) represents the images of the sets B and X under f right?