1549901369-Elements_of_Real_Analysis__Denlinger_

B.l Sets and the Algebra of Sets 615

These sets are conveniently illustrated in the following "Venn diagrams:"

(a)A u B (b)A n B

(c) Ac (d) B-A

Figure B.1

Examples B.1.5 Let U ={real numbers}. Then

(a) [O, 3] U [l, 5] = [O, 5]

(b) [0,3] n [1,5] = [1,3]

(c) [O, W = (-oo,O) u (3, +oo)

(d) [O, 3] - [l, 5] = [O, 1)

(e) [l , 5] - [O, 3] = (3, 5] D

Definition B.1.6 We say that A~ B (A is a subset of B) iff every element
of A is also an element of B.

For example, {l, 2, 3} ~ N and {x: x^2 - 3x + 2 = O} ~ N, but [l, 3] ~ N.

Theorem B.1.7 (Algebra of Sets) For any sets A, B, CE U ,

(a) A= B {::}A ~ B and B ~ A.

(b) 0 ~A, A~ A, and A~ U.

( c) A n B ~ A and A n B ~ B.

(d) A~AUB andB~AUB.

(e) AU B =A iff B ~A.

(f) An B =A iff A~ B.