PROOF:
How do we know that ?
And how does that mean that ?
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.
PROOF:
How do we know that ?
And how does that mean that ?
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.
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".
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.