1. Prove or disprove each of the following statements: (a) Va, be Q, ab is rational....

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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 √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 <d, then ac < bd. *(e) For any nonempty sets A, B and C, A\ (B\C) = (A \ B) \ C. (f) For any nonempty sets A, B and C, if Ax BC Ax C, then BCC. *(g) For any sets A and B, P(AUB) = P(A) UP(B). (h) There exists n € Z such that n² + 2 = 0 (mod 4). *(i) There exist a, b = Z and there exists n N with n ≥ 2 such that a ‡ b (mod n) and a² = b² (mod n). *(j) For any a, b € Z and for any m, n E N with m, n ≥ 2, if a = b (mod m) and a = b (mod n), then a = b (mod mn). *(k) For any n E N with n ≥ 3, if 2 - 1 is prime, then n is odd. *(1) There exist distinct irrational numbers a and b such that a³ is rational. (m) Every odd integer is the sum of three odd integers. (n) VER, Ey R, e² - x > 0. (o) y ER, Vr € R, e² - x > 0. (p) VrZ, Vy = Z, 32 € Z, z² + 2xz - y² = 0. Mathematical Induction 2. (Sum of a Geometric Sequence) Prove that for any integer n ≥ 0 and any real number r # 1, we have p²+1 1+r+²+ 21 +r" = for any n € N. -1 r-1 3. (Generalization of De Morgan's Law) Prove that for any subsets A₁, A2,..., A₁, of a universal set U and for any integer n ≥ 2, = AU AU... UA (A₁4₂.A₂) 6. Prove that for any q ER with 0 <q < 1/2, *4. Prove that 17" - 10" is divisible by 7 for any positive integer n. 5. Prove that for any n € N, we have . 4" > n³. (Hint: Show that the inequality is true for n = 1 and use induction for n ≥ 2.) (1+q)" ≤ 1+2"q *7. Prove that for any n E N. 1 1 1 ī+ ₂ + 3 + Prove that *8. Consider the sequence F₁, F2, F3,... of Fibonacci numbers satisfying F₁ = 1, F₂ = 1 and F₁ = Fn-1 + Fn-2 for any n ≥ 3. -15 21+2 F+F++F FFn+1 for any n € N. then ₁2" + 1 for all n E N. Strong Form of Induction 9. Prove that if a sequence 21, 22, 23,... of numbers satisfying 2₁ = 3 and 22 = 5 and In = 3xn-1-2In-2 for any integer n > 3, 10. A sequence ₁, 2, 3,... of numbers is defined by ₁ = 1, ₂ = 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 F₁, F2, F3,.... be the sequence of Fibonacci numbers (defined in Q.8). Prove that for any n € N, F₁, 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 P₁, 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.) 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 <d, then ac < bd. *(e) For any nonempty sets A, B and C, A\ (B\C) = (A \ B) \ C. (f) For any nonempty sets A, B and C, if Ax BC Ax C, then BCC. *(g) For any sets A and B, P(AUB) = P(A) UP(B). (h) There exists n € Z such that n² + 2 = 0 (mod 4). *(i) There exist a, b = Z and there exists n N with n ≥ 2 such that a ‡ b (mod n) and a² = b² (mod n). *(j) For any a, b € Z and for any m, n E N with m, n ≥ 2, if a = b (mod m) and a = b (mod n), then a = b (mod mn). *(k) For any n E N with n ≥ 3, if 2 - 1 is prime, then n is odd. *(1) There exist distinct irrational numbers a and b such that a³ is rational. (m) Every odd integer is the sum of three odd integers. (n) VER, Ey R, e² - x > 0. (o) y ER, Vr € R, e² - x > 0. (p) VrZ, Vy = Z, 32 € Z, z² + 2xz - y² = 0. Mathematical Induction 2. (Sum of a Geometric Sequence) Prove that for any integer n ≥ 0 and any real number r # 1, we have p²+1 1+r+²+ 21 +r" = for any n € N. -1 r-1 3. (Generalization of De Morgan's Law) Prove that for any subsets A₁, A2,..., A₁, of a universal set U and for any integer n ≥ 2, = AU AU... UA (A₁4₂.A₂) 6. Prove that for any q ER with 0 <q < 1/2, *4. Prove that 17" - 10" is divisible by 7 for any positive integer n. 5. Prove that for any n € N, we have . 4" > n³. (Hint: Show that the inequality is true for n = 1 and use induction for n ≥ 2.) (1+q)" ≤ 1+2"q *7. Prove that for any n E N. 1 1 1 ī+ ₂ + 3 + Prove that *8. Consider the sequence F₁, F2, F3,... of Fibonacci numbers satisfying F₁ = 1, F₂ = 1 and F₁ = Fn-1 + Fn-2 for any n ≥ 3. -15 21+2 F+F++F FFn+1 for any n € N. then ₁2" + 1 for all n E N. Strong Form of Induction 9. Prove that if a sequence 21, 22, 23,... of numbers satisfying 2₁ = 3 and 22 = 5 and In = 3xn-1-2In-2 for any integer n > 3, 10. A sequence ₁, 2, 3,... of numbers is defined by ₁ = 1, ₂ = 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 F₁, F2, F3,.... be the sequence of Fibonacci numbers (defined in Q.8). Prove that for any n € N, F₁, 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 P₁, 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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