2. Q1. Write the following sets?
1. x ∈ Z | x = y 2 f or some integer y ≤ 3
2. x ∈ Z | x2 = y f or some integer y ≤ 3
Q2. Suppose the universal set U = {−1, 0, 1, 2}, A = {0, 1, 2} and B = {−1, 2}. What are the following sets?
1. A ∩ B
2. A ∪ B
3. Ac
4. A − B
5. A × B
6. P (B)
Q3. Prove the following by Induction
1. 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 −1
3. r = 1, ∀n ≥ 1, 1 + r + ..... + rn = r−1
2
n∗(n+1)
4. ∀n ≥ 1, 14 + 23 + ..... + n3 = 2
5. ∀n ≥ 1, 22n − 1isdivisibleby3
6. ∀n ≥ 2, n3 − nisdivisibleby6
7. ∀n ≥ 3, 2n + 1 < 2n
1 1 1 √
8. ∀n ≥ 2, √
1
+ √
2
+ .... + √
n
> n
Q4. Prove the following by Contradiction
1. There exists no integers x and y such that 18x + 6y = 1
2. If x, y ∈ Z, then x2 − 4y − 3 = 0.
Q5. Prove the following by Contrapositive
1. ∀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 < 0
4. 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)
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