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1. Prove or disprove each of the following statements: (a) Va, be Q, ab is rational. (b) There exists distinct positive integers such that

1. Prove or disprove each of the following statements: (a) Va, be Q, ab is rational. (b) There exists distinct positive integers such that a+b = a+b-2. (c) For any set A, if A\B=0 for any set B, then A = 0). *(d) For any a, b, c, d e R, if a < b and c 0. (o) y ER, Vr R, e - x > 0. (p) VrZ, Vy = Z, 32 Z, z + 2xz - y = 0. Mathematical Induction 2. (Sum of a Geometric Sequence) Prove that for any integer n 0 and any real number r # 1, we have p+1 1+r++ 21 +r" = for any n N. -1 r-1 3. (Generalization of De Morgan's Law) Prove that for any subsets A, A2,..., A, of a universal set U and for any integer n 2, = AU AU... UA (A4.A) 6. Prove that for any q ER with 0 *7. Prove that for any n E N. 1 1 1 + + 3 + Prove that *8. Consider the sequence F, F2, F3,... of Fibonacci numbers satisfying F = 1, F = 1 and F = Fn-1 + Fn-2 for any n 3. -15 21+2 F+F++F FFn+1 for any n N. then 2" + 1 for all n E N. Strong Form of Induction 9. Prove that if a sequence 21, 22, 23,... of numbers satisfying 2 = 3 and 22 = 5 and In = 3xn-1-2In-2 for any integer n > 3, 10. A sequence , 2, 3,... of numbers is defined by = 1, = 4, 13 In = In-1-n-2+In-3+2(2n-3) for any integer n 4. Guess a formula for , and prove that your guess is correct. 3 = 9 and *11. Let F, F2, F3,.... be the sequence of Fibonacci numbers (defined in Q.8). Prove that for any n N, F, is even if and only if n = 0 (mod 3). *12. (Fundamental Theorem of Arithmetic) Prove that every integer n 2 can be written as a product of one or more (not necessarily distinct) primes, that is, n= pip2 p, for some prime numbers P, P2, Pr. (Hint: In the inductive step, if k + 1 is prime, then we are done. Otherwise, you may need to use the following definition of composite numbers: An integer n 2 is called composite if it can be factored as n = ab for some integers a and b satisfying 2 a, b < n.)

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