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= = = = Problem 25: Let P and Q be two predicates, and let their truth sets be denoted as Tp,TQ, and falsity sets be denoted as Fp, FQ. Prove the following identities. 1. Tp n TQ = TPAQ 2. Tp UTQ = TpvQ 3. Fp FQ = FPVQ 4. Fp UFQ = FPAQ Problem 26: What are the most general conditions on the truth sets Tp and TQ making 1. (P(2) Q(x)) = P(1) true 2. (P3) Q(2)) = Q(2) true Problem 27: Suppose the domain D for predicates P, Q, and S is {a,b,c}. Express the following propositions without using quantifiers. - 1. Vr P(1) 2. (Vx R(1) (Ix S(x)) Problem 23: Let Dx :=R. Which of the following are predicates? (1) 12+1