# Mathematics Assignment | Custom Assignment Help

1. (4 marks) Prove that for all sets A, B, and C,

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A × (B ∩ C) = (A × B) ∩ (A × C).

2. (5 marks) Consider the function f : Z×Z → Z where f((x, y)) = 3x+ 5y for all (x, y) ∈ Z×Z.

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3. (8 marks) Let S = {(a1, a2, . . . , an)| n ≥ 1, ai ∈ Z

≥0

for i = 1, 2, . . . , n, an 6= 0}. So S is the set

of all finite ordered n-tuples of nonnegative integers where the last coordinate is not 0. Find a

bijection from S to Z

+.

4. (6 marks) Define a relation r on R as follows. For all a, b ∈ R, a r b if and only if abbaab < 0.

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