1. ## a question about set relation between functions

hi

I have few doubts about one set theorem which I proved.

Suppose that $f:X\rightarrow Y$ with $A\subseteq X$
and $B\subseteq Y$ . I had to prove that

$A\subseteq f^{-1}(f(A))$

I did prove that but I have some questions. Shouldn't there be an equality

$A= f^{-1}(f(A))$ because when we talk about the inverse functions, we talk about one-to-one functions. So I can't think of situation
where

$\textstyle A\subset f^{-1}(f(A))$

Can you ? and question related to latex typesetting. how do I reduce font size ?

thanks

2. Originally Posted by issacnewton
hi

I have few doubts about one set theorem which I proved.

Suppose that $f:X\rightarrow Y$ with $A\subseteq X$
and $B\subseteq Y$ . I had to prove that

$A\subseteq f^{-1}(f(A))$

I did prove that but I have some questions. Shouldn't there be an equality

$A= f^{-1}(f(A))$ because when we talk about the inverse functions, we talk about one-to-one functions. So I can't think of situation
where

$\textstyle A\subset f^{-1}(f(A))$

Can you ? and question related to latex typesetting. how do I reduce font size ?

thanks

In this case $f^{-1}(A)$ does not denote the inverse function of f applying on a set A, but the

inverse image of A under ANY function f. Two very different concepts. For example

$f:\mathbb{R}\rightarrow \mathbb{R}_+\,,\,\,f(x):=x^2\,,\,\,A=[-1,1)\Longrightarrow f(A)=[0,1]\Longrightarrow A\subset f^{-1}(f(A))=[-1,1]$

Tonio

3. thanks tonio

that makes sense