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  • 1. Assignment 3Deadline: April 26 (Thursday Class), April 27(Friday Class) Harshit Kumar April 13, 2012
  • 2. Q1. Write the following sets?1. x ∈ Z | x = y 2 f or some integer y ≤ 32. x ∈ Z | x2 = y f or some integer y ≤ 3Q2. Suppose the universal set U = {−1, 0, 1, 2}, A = {0, 1, 2} and B = {−1, 2}. What are the following sets?1. A ∩ B2. A ∪ B3. Ac4. A − B5. A × B6. P (B)Q3. Prove the following by Induction1. If a is odd and b is odd, then a ∗ b is odd.2. Any integer i > 1 is divisible by p, where p is a prime number. r n+1 −13. r = 1, ∀n ≥ 1, 1 + r + ..... + rn = r−1 2 n∗(n+1)4. ∀n ≥ 1, 14 + 23 + ..... + n3 = 25. ∀n ≥ 1, 22n − 1isdivisibleby36. ∀n ≥ 2, n3 − nisdivisibleby67. ∀n ≥ 3, 2n + 1 < 2n 1 1 1 √8. ∀n ≥ 2, √ 1 + √ 2 + .... + √ n > nQ4. Prove the following by Contradiction1. There exists no integers x and y such that 18x + 6y = 12. If x, y ∈ Z, then x2 − 4y − 3 = 0.Q5. Prove the following by Contrapositive1. ∀n ∈ Z, if nk is even, then n is odd.2. ∀x, y ∈ Z, if x2 (y 2 − 2y) is odd, then x and y are odd.3. ∀x ∈ R, if x2 + 5x < 0, then x < 04. If n is odd, then (n2 − 1) is divisible by 8.5. If n ∈ N and 2n − 1 is prime, then n is prime.6. ∀x, y ∈ Z and ninN, if x3 ≡ y 3 (mod n) 1