Question

1. Write the following sets in list form. (For example, {x | x ∈N,1 ≤ x < 6} would be {1,2,3,4,5}.) (a) {a | a ∈Z,a2 ≤ 1}. (b) {b2 | b ∈Z,−2 ≤ b ≤ 2} (c) {c | c2 −4c−5 = 0}. (d) {d | d ∈R,d2 < 0}.

2. Let S be the set {1,2,{1,3},{2}}. Answer true or false: (a) 1 ∈ S. (b) {2}⊆ S. (c) 3 ∈ S. (d) {1,3}∈ S. (e) {1,2}∈ S (f) {2}∈ S. (g) {2,{2}}⊆ S. (h) {3}⊆ S. (i) ∅∈ S. (j) ∅⊆ S.

3. Let A = N, B = {y3 | y ∈Z}, C = {z + 3 | z ∈Z, −2 ≤ z ≤ 10}, D = Q. For each of the following, state whether they are true or false. If true, give a reason; if false, give a counterexample. (a) A ⊆ B. (b) B ⊆ A. (c) C ⊆ A. (d) A∪B ⊆ D. (e) C ∩D ⊆ B.

4. For each of the following, give an example of a statment P and a statement Q that satisfy the given conditions. (a) P ⇒ Q but Q 6⇒ P. (b) P if and only if Q. (c) P if Q.

5. Prove that if x is an odd integer, then there exists some integer y such that x2 = 4y +1

Answer 1