618 Appendix B • Sets and FunctionsPart 1: Suppose x EB - ( n A;). Then x EB but x ~ ( n A;). That
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is , x EB but rv \:/)...EA, x EA;. By "quantifier negation,''^5 this means x EB
but 3)... EA 3 x ~A;. That is, x EB but 3)... EA 3 x E Ai. Equivalently,
3)... EA 3 x EB n Ai. Equivalently, 3)... EA 3 x EB -A;_. But that means
x E LJ (B - A;). Therefore, B - ( n A;) <:;:; LJ (B - A;).
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Part 2: Suppose x E LJ (B - A;). Then 3)... EA 3 x EB - A;. That is ,
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3 >. E A 3 x E B n Ai· Equivalently, x E B but 3 )... E A 3 x E Ai· By quantifier
negation, this means x E B but rv \:/)... E A, x E A;. That is, x E B but
x ~ ( n A;_); i.e., x EB-( n A;_). Therefore, LJ (B-A;_) <:;:; LJ (B-A;_).
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By Parts 1, 2, and Theorem B .1.7 (a), B - ( n A;_) = LJ (B -A;_). •
.AEA .AEAEXERCISE SET B.1- In each of the following, a universal set U and sets A, B, and C are given.
Find AnB, AUE, N, Be, A-B, B-A, AU(BnC), and An(BUC).
(a) U = {1,2,3,-·· ,10}, A= {1,2,3,4,5}, B = {4,5,6,7}, and C =
{3, 4, 5}.
(b) U = {1, 2, 3, · · · , 10}, A = {1, 2, 3}, B = { 4, 5, 6}, and C = {2, 4, 6, 8, 10}.
(c) U ={all real numbers}, A= (0,4), B = [3,6], C = (2,5).
(d) U ={all real numbers}, A= (-oo, 2), B = [1, +oo), C = (-1, 1). - Prove Theorem B.1.7 (e).
- Prove Theorem B.1.7 (h).
- Prove Theorem B.1.7 (k).
- Finish the proof of Theorem B.1.7 (o) by proving "Part 2."
- Prove Theorem B.1.7 (p).
- In each of the following, a collection of sets {A; : )... E A} is given. Assume
U ={all real numbers}. Find n A;, LJ A;, LJ A1, and LJ A1. In each
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case, verify Theorem B .1.10 .(a) or (b).
(a) {A;:)... EA}= {(-n,n): n EN}.
(b) {A;:)... EA}= {(-oo,n): n EN}.
(c) {A;:>. EA}= { (-~, ~) : n EN}. - See section A.2.