# Sets and functions

• September 12th 2009, 10:33 AM
abc512
Sets and functions

Prove that if $f$ is a function, $f^{-1}[A \cap B] = f^{-1}[A] \cap f^{-1} [b]$.

Also, prove that if $f$ is a function, $f^{-1}[A - B] = f^{-1}[A] - f^{-1} [b]$.

All of the B's should be capitalized, I'm not sure why it isn't coming out that way.
• September 12th 2009, 11:14 AM
Taluivren
$x\in f^{-1}[A \cap B]$
$f(x) \in A \cap B$
$f(x) \in A$ and $f(x) \in B$
$x\in f^{-1}[A]$ and $x\in f^{-1}\left[B\right]$
$x \in f^{-1}[A] \cap f^{-1}\left[B\right]$

check that each pair of consecutive lines is a pair of equivalent statements.

$x\in f^{-1}[A - B]$
$f(x) \in A - B$
$f(x) \in A$ and $f(x) \not \in B$
$x\in f^{-1}[A]$ and $x\not \in f^{-1}\left[ B \right]$
$x \in f^{-1}[A] -f^{-1}\left[B\right]$

again check each pair of consecutive lines is a pair of equivalent statemetns.

I am having the same problem with [b] as you, i tried \left[ B \right] and it worked.