Answer Happy

. 3. Let x,y E Z. Prove that (x + 1)y2 is even if and only if x is odd or y is even. 4. Let x,y e Z. Prove that if xy and x + y are even, then both x and y are even. 5. Prove that for every integer x, the integers 3x + 1 and 5x + 2 are of opposite parity. 6. An integer x is called a Moffett number provided x has a remainder of 2 when divided by 4. Write a formal proof of the statement "If x is not a Moffett number, then x2 – 1 is not a Moffett number." 7. An integer x is called a Sperry number provided x has an odd remainder when divided by 4. Write a formal proof of the statement "lfx and y are not Sperry numbers, then their product is not a Sperry number." 8. Let A, B, and C be sets. Prove that (A - B) n(A-C) = A -(BUC). sets 9. Let A, B, and C be sets. Prove that if An B = An C and AU B = AU C, then B = C. 10. Let A = {n € Zn = 1 (mod 2)) and B = {n e Zin = 3 (mod 4)). Prove B SA. 11. Let A, B, and C be sets. Prove that A CB O C if and only if A C B or A SC. 12. Prove that if a 2 2 and n 2 1 are integers such that a2 + 1 = 2", then a is odd. 13. A relation R is defined on Z by arb if and only if 3a + 5b = 0 (mod 8). Prove that R is an equivalence relation 14. Prove that the function f RR defined by f(x) = 2x + 3 is one-to-one and onto, relations 15. Prove that the function f: RR defined by f(x) = -3x +5 is one-to-one and onto. 16. Prove that 13 + 23 +33 +...+n3 for every positive integer n. Induction - (+1)