set proof

• Dec 18th 2008, 06:53 AM
qwerty321
set proof
Hello

Let
A, 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
• Dec 18th 2008, 07:22 AM
Isomorphism
Quote:

Originally Posted by qwerty321
Hello

Let
A, 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.
• Dec 18th 2008, 07:45 AM
qwerty321
when you pu the little c above A or B, what do u mean?
and for the contrapositivity how did u pass from x belongs to A^cUC to A^cUBUC?

thank you
• Dec 18th 2008, 07:50 AM
Isomorphism
Quote:

Originally Posted by qwerty321
when you pu the little c above A or B, what do u mean?
and for the contrapositivity how did u pass from x belongs to A^cUC to A^cUBUC?

thank you

A^c stands for A complement.

Since $\displaystyle x \in C, x \in A^c \cup B \cup C$.