# Thread: Help understanding this proof: E is a subset of f^(-1)(f(E))

1. ## Help understanding this proof: E is a subset of f^(-1)(f(E))

$\displaystyle \text{Let } f: X\rightarrow Y, \text{ where } X \text{ and } Y \text{ are sets.}$

$\displaystyle \text{Prove that }E \subseteq f^{-1}(f(E)) \text{ } \forall E \subseteq X.$

PROOF:
$\displaystyle \text{Let } x \in E.$
$\displaystyle \text{Then, }y = f(x) \in f(E), \text{ so } x \in f^{-1}(f(E)).$
$\displaystyle \text{ Hence } E \subseteq f^{-1}(f(E))$

How do we know that $\displaystyle y = f(x) \in f(E)$?

And how does that mean that $\displaystyle x \in f^{-1}(f(E))$?

I guess I'm struggling with comprehending how a function is defined. We're using the book called Mathematical Analysis by Apostol and it's not an easy read for me.

2. ## Re: Help understanding this proof: E is a subset of f^(-1)(f(E))

Hey amthomasjr.

A function is defined as a mapping from one set to another where the mapping is one to one [often known as bijective].

The "funny" e sign means "is an element of" which means if you have a collection of "things" then there is an element inside that collection that satisfies some "criterion".

3. ## Re: Help understanding this proof: E is a subset of f^(-1)(f(E))

Originally Posted by amthomasjr
$\displaystyle \text{Let } f: X\rightarrow Y, \text{ where } X \text{ and } Y \text{ are sets.}$
$\displaystyle \text{Prove that }E \subseteq f^{-1}(f(E)) \text{ } \forall E \subseteq X.$
PROOF:
$\displaystyle \text{Let } x \in E.$
$\displaystyle \text{Then, }y = f(x) \in f(E), \text{ so } x \in f^{-1}(f(E)).$
$\displaystyle \text{ Hence } E \subseteq f^{-1}(f(E))$
How do we know that $\displaystyle y = f(x) \in f(E)$?
It is really knowing the definitions,
If $f:X\to Y$ and $E\subset X$, then by definition $\large f(E)=\{f(x) :x\in E\}$
$f:X\to Y$ and $H\subset Y$, then by definition $\large f^{-1}(H)=\{t\in X :f(t)\in H\}$

4. ## Re: Help understanding this proof: E is a subset of f^(-1)(f(E))

Originally Posted by Plato
It is really knowing the definitions,
If $f:X\to Y$ and $E\subset X$, then by definition $\large f(E)=\{f(x) :x\in E\}$
$f:X\to Y$ and $H\subset Y$, then by definition $\large f^{-1}(H)=\{t\in X :f(t)\in H\}$
So I just let $\displaystyle f(E) = H$ since $\displaystyle f(E) \subseteq Y$and apply it to the second definition. That makes sense.

5. ## Re: Help understanding this proof: E is a subset of f^(-1)(f(E))

Originally Posted by amthomasjr
So I just let $\displaystyle f(E) = H$ since $\displaystyle f(E) \subseteq Y$and apply it to the second definition.
From these
If $f:X\to Y$ and $E\subset X$, then by definition $\large f(E)=\{f(x) :x\in E\}$
$f:X\to Y$ and $H\subset Y$, then by definition $\large f^{-1}(H)=\{t\in X :f(t)\in H\}$

We know that $f(E)\subset Y$ thus $f^{-1}(f(E))\subset X$.

If $t\in E$ then $f(t)\in f(E)$ so by definition $t\in f^{-1}(f(E))$ by definition.

6. ## Re: Help understanding this proof: E is a subset of f^(-1)(f(E))

Originally Posted by Plato
From these
If $t\in E$ then $f(t)\in f(E)$ so by definition $t\in f^{-1}(f(E))$ by definition.
Suppose I was only given $\displaystyle$f(t)\in f(E)$$. Is deducing that \displaystyle t \in E valid? I know in some cases the converse of a conditional isn't logically equivalent. 7. ## Re: Help understanding this proof: E is a subset of f^(-1)(f(E)) Originally Posted by amthomasjr Suppose I was only given \displaystyle f(t)\in f(E)$$. Is deducing that $\displaystyle t \in E$ valid?
No That is not valid.

BUT in the OP you were told to show that $\large E\subset f^{-1}(f(E)).$