# Math Help - Proofs

1. ## Proofs

1.
Prove: For all sets A and B, if A is a subset of B, then not B is a subset of not A.

2.
Prove: For all sets A,B,and C, A - (B U C) = (A-B) and (A-C)

2. #1 is really only about logic. Recall that $\left( {P \to Q} \right) \equiv \left( {\neg Q \to \neg P} \right)$.
Thus “If x is in A then x is in B.” is equivalent to “If xi is not in B then x is not in A”.

#2
$\begin{gathered}
A\backslash \left( {B \cup C} \right) \equiv A \cap \left( {B \cup C} \right)^c \hfill \\
\equiv A \cap \left( {B^c \cap C^c } \right) \hfill \\
\equiv \left( {A \cap B^c } \right) \cap \left( {A \cap C^c } \right) \hfill \\
\equiv \left( {A\backslash B} \right) \cap \left( {A \backslash C} \right) \hfill \\
\end{gathered}$