
Sets and functions
Any ideas about these?
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.

$\displaystyle x\in f^{1}[A \cap B]$
$\displaystyle f(x) \in A \cap B$
$\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.