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Q7- 6 points Consider the following statement: For all sets A and B, (A ∪ B) ∩ C  =  A ∪ (B ∩ C) Write a sentence that describes what would be required to show that this statement is false, and find subsets of {1, 2, 3, 4, 5, 6} which can be used to meet that requirement.

Q6- 8 points Write a proof for the following statement: For all sets A and B, if A ⊆ B, then A ∪ B ⊆ B.

Q3 – 6 points Let Z be the set of all integers and let  A0 = {n ∈ Z | n = 4k, for some integer k} A1 = {n ∈ Z | n = 4k + 1, for some integer k}  and A2 = {n ∈ Z | n = 4k + 2, for some integer k}. Is {A0, A1, A2}  a partition of Z? Explain why or why not?

Q5 – 6 points Let A = {p, q, r}, B = {1, 2, 4}, and C = {0, 4}. Use set-roster notation to write the following set: A × (B ∩ C)

Q2- 9 points Let G = {1, 2, 3} and  H = {4, 6, 9} and define a relation R from G to H as follows: For every (x, y) ∈ G × H,  (x, y) ∈ R  means that  (x + y)/2  is an integer. Is 2 R 6? Is (2, 9) ∈ R? Write R  as a set of ordered pairs.

Q9- 6 points Use an element argument to prove the following statement: For every set A,  A ∩ ∅ = ∅.

Q15 – 6 points Define a function f : Z → Z by the rule f(n) = 2n , for every integer n. Is f an onto function? Is f a one-to-one function? Prove using the definitions of onto and one-to-one.

Q2 – 9 points Let G = {1, 2, 3} and  H = {5, 6, 7} and define a relation R from G to H as follows: For every (x, y) ∈ G × H,  (x, y) ∈ R  means that  (x – y)/2  is an integer. Is 3 R 6? Is (2, 7) ∈ R? Write R  as a set of ordered pairs.  

Q13 – 6 points Define a function F : Z × Z → Z × Z as follows: F(x, y) = (y−1, 3−x) for all (x, y) in Z × Z.  Find F(0, 0) and F(3, 4).

Q6 – 8 points Write a proof for the following statement: For all sets A and B, if A ⊆ B, then A ⊆ A ∩ B.