156 METHODS OF MATHEMATICAL PROOF, PART I Chapter 5Throughout Exercises 3 through 6, let all sets involved be subsets of a universal
set U. Use properties of sets proved in Article 4.1 and an argument by transitivity
in each case.- Prove that for any sets A and B
(a) A-@=A (b) 0-A=@
(c) A A 0 *(d) A A U = A'
(e) AAA=0
(f) If B = 0, then A = (A n B') u (A' n B)
(g) If B = A, then U = (A' u B) n (A u B'). [Note: The converses of the results
in (f) and (g) are also true; see Example 8 and Exercise l(h), Article 6.2,
respectively.] - Prove that for any sets A and B:
(a) A = (A u B) n (A u B')
(b) A = (A n B) u (A n B')
(c) (A n B) u (A' n B) u (A n B') u (A' n B') = U
(d) (A u B) n (A' u B) n (A u B') n (A' u B') = 0
(e) AAB=BAA - (a) Prove that, for any sets A, B, and C:
(i) A-(BuC)=(A-B)n(A-C)
*(ii) (Au B)-C=(A-C)u(B-C)
(iii) (A - B) - C = (A - C) - (B - C)
(b) Prove that, for any sets A, B, C, and D:
(i) An(BuCuD)=(An B)u(AnC)u(An D)
(ii) A u (B n C n D) = (A u B) n (A u C) n (A u D). (Hint: Use associativ-
ity. These results generalize distributivity to "distributivity across three
sets." In Article 5.4, on mathematical induction, we will see how to prove
, "distributivity across any finite number of sets.") - Prove that for any set A and for any collection {Bi 1 i = 1,2,.. .} of sets indexed
by N (recall Exercise 4, Article 4.2): - Use the definition of (;) from Article 1.5, together with the facts that (n + I)! =
(n + l)n! (for all n E N) and O! = 1, to show that whenever each expression is de-
fined, the given equation must hold:
*(a) ("3 =(A) + 6) -
(b) 6: 1) = [(n - k)l(k + l)l(;)
(c) a(?) = @GI:) (Suggestion: Try some specific substitutions before attempt-
ing the proof.) - The absolute value of a real number x, denoted 1x1, is defined by the rule
It follows directly from the definition that - 1x1 ( x I 1x1 for all x E R. From this