Exercise 1.2.5 (De Morgan's Laws). Let

A

and

B

be subsets of

R

.\ (a) If

xin(A\\\\cap B)^(c)

, explain why

xinA^(c)\\\\cup B^(c)

. This shows that

(A\\\\cap B)^(c)sube

A^(c)\\\\cup B^(c)

.\ (b) Prove the reverse inclusion

(A\\\\cap B)^(c)supeA^(c)\\\\cup B^(c)

, and conclude that

(A\\\\cap B)^(c)=A^(c)\\\\cup B^(c)

.

Exercise 1.2.5 (De Morgan's Laws). Let A and B be subsets of R. (a) If x(AB)c, explain why xAcB)c. This shows that (AB)c AcBc. 12 Chapter 1. The Real Numbers (b) Prove the reverse inclusion (AB)cAcBc, and conclude that (AB)c=AcBc