Answer Happy

a) Prove logical equivalence: (p+q) = (p^g) ^ (q^p) Q ~P ~Q PQ Q^~P~(P^~Q) ~(Q^~P) (P <=> Q) ~(P^~Q)^~(Q^~P) 1 1 0 1 1 1 0 0 0 1 0 1 1 0 0 1 1 0 0 1 1 Hence logically equivalent b) Translate these statements into English, where C (x) is "x is a comedian" and F(x) is "x is funny" and the domain consists of all people. ii) x(c(x) ^ F(x)) 1) Vx(C(x) → F(x)) iii) Ex(C(x) → F(x)) iv) Ex(C(x)^F(x))/ Every comedian is funny ii) Every person is funny comedian iii) There is a person such that if he/she is a comedian, then he/she is funny. ix) Some comedians are funny c) Show that if n is an integer and n³ + 5 is odd, than n is even using proof by Contradiction d) Prove, by induction that 1 1-2+2-3+...+ n(n+1)== n(n+1)(n+2) for all n ≥1 P OT OH 0 1 0 OTO OOO 0 1 0 0 OOH 0 0 OFF 100 1001