Hello
LetA, B and C be 3 sets. Prove that:
if x belongs to (A-B)-C then x belongs to A-C using:
a)direct proof.
b)proof by contraposition
thank you
a) $\displaystyle x \in A \cap B^c \cap C^c \Rightarrow x \in A, x \notin B, x \notin C \Rightarrow x \in A, x \notin C \Rightarrow x \in A$\$\displaystyle C$
b) $\displaystyle x \notin A$\$\displaystyle C \Rightarrow x \notin A, x \in C \Rightarrow x \in A^c \cup C \Rightarrow x \in A^c \cup B \cup C \Rightarrow x \in (A \cap B^c \cap C^c)^c \Rightarrow x \notin (A$\$\displaystyle B)$\$\displaystyle C$
Thus $\displaystyle x \notin A$\$\displaystyle C\Rightarrow x \notin (A$\$\displaystyle B)$\$\displaystyle C$
Which is proof of the contrapositive.