1. ## Sets and functions

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

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

All of the B's should be capitalized, I'm not sure why it isn't coming out that way.

2. $\displaystyle x\in f^{-1}[A \cap B]$
$\displaystyle f(x) \in A \cap B$
$\displaystyle f(x) \in A$ and $\displaystyle f(x) \in B$
$\displaystyle x\in f^{-1}[A]$ and $\displaystyle x\in f^{-1}\left[B\right]$
$\displaystyle x \in f^{-1}[A] \cap f^{-1}\left[B\right]$

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

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

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.