EXAMPLE 4.4.7. Suppose 1) C C R, and 2) V1: 6 R, ((332 = 9) => (33 E 0)). Show EIC E R,C E C. PROOF. Let C = 3 E R. Then (:2 = 32 = 9, and letting 33 = c in Hypothesis (2) tells us that (02 = 9) => ((2 E 0). Therefore 0 E C. EXERCISES 4.4.9. 1) Assume Vzv E R, (:32 E Z). Show 16 E Z. 2) Assume A C B and A a Q. Show B 7 I2). 3) Assume (a) for every :13 E A, either w E B, or 2: 0. Show B # Q). EXERCISES 4.5.4. Suppose A and B are sets. 1) Show A \\ B = AnB. 2) Show A = (A \\ B) U (An B). 3) Prove De Morgan's Laws: (a) A = A. (b) AnB = AUB. (c) AUB = AnB. 4) Show that if A = B, then A = B. [Hint: Follows immediately from one of De Morgan's Laws.] EXERCISES 4.5.5. Suppose A, B, and C are sets. 1) Show that A is disjoint from B if and only if A C B. 2) Show A B is disjoint from B. 3) Show that if A is disjoint from B, and C is a subset of B, then A is disjoint from C. 4) Show that A \\ B is disjoint from An B. 5) Show that A is disjoint from BUC iff A is disjoint from both B and C. EXERCISES 4.5.6. 1) Show AUB = (A \\ B)U(B A) U(AnB). 2) Show the three sets A \\ B, B . A, and An B are all disjoint from each other. Recall from Remark 3.3.20(2) that sets A1, A2, ..., An are pairwise-disjoint iff A; is disjoint from Aj whenever i * j.EXERCISE 4.5.7. Suppose the sets A1,A2, . . . ,An are pairwisedisjoint. Show: 1) The sets A1, A2, . . . ,An_1 are pairwisedisjoint, if n > 1. 2) A\" is disjoint from A1 U A2 U - - - U An_1, if n > 1. EXERCISES 4.6.5. Explain how you know that each of the following deductions is not valid. 1) 1x, (x E A), 1x, (x E B), .. Fx, ((x EA) & (xEB)) 2) Va E A, bE B, (at b), AFO, .. VbEB, Ha E A, (a * b). 3) A # B, . . A UB * A. 4) Vr E A, (x ( B), Vx EB, (x tA), . AFB. EXERCISES 4.6.6. Explain how you know that each of these deductions is not valid. 1 AUB C EUF, . AnB C EnF. 2) ACB, X CY, . A \\ X C B Y. 3) AnB + 0, BCC, .: A CC. 4 ) 1x , ((EP ) & (at Q ) ) , .. Va , ((xep ) = (x*Q ) ) . (( tac y Ex)