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)

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- Dec 11th 2007, 10:12 PMscottyflamingoProofs
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) - Dec 12th 2007, 02:19 AMPlato
#1 is really only about logic. Recall that $\displaystyle \left( {P \to Q} \right) \equiv \left( {\neg Q \to \neg P} \right)$.

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

$\displaystyle \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} $