Exam (elaborations) TEST BANK FOR Topology 2nd Edition by James Munkres (Solution Manual)

Chapter 1 Set Theory and Logic x1 Fundamental Concepts Exercise 1.1 Check the distributive laws for [ and and DeMorgan's laws. Solution: Suppose that A, B, and C are sets. First we show that A (B [ C) = (A B) [ (A C). Proof. We show this as a series of logical equivalences: x 2 A (B [ C) , x 2 A ^ x 2 B [ C , x 2 A ^ (x 2 B _ x 2 C) , (x 2 A ^ x 2 B) _ (x 2 A ^ x 2 C) , x 2 A B _ x 2 A C , x 2 (A B) [ (A C) ; which of course shows the desired result. Next we show that A [ (B C) = (A [ B) (A [ C). Proof. We show this in the same way: x 2 A [ (B C) , x 2 A _ x 2 B C , x 2 A _ (x 2 B ^ x 2 C) , (x 2 A _ x 2 B) ^ (x 2 A _ x 2 C) , x 2 A [ B ^ x 2 A [ C , x 2 (A [ B) (A [ C) ; which of course shows the desired result. Now we show the rst DeMorgan's law that A