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Math 220 Assignment 4 Due October 16th Throughout, let A, B, C and D be sets, and let f : D → C be a function. In particular, you no longer need to include these sentences in your proofs. 1. Which of the following functions are surjective? Give a formal proof. In each case, the domain and codomain are both Z, the set of integers. (a) f (n) = 2n + 1 n/3 n is 3-divisible (b) f (n) = n − (n%3) otherwise n − 1 n is even (c) f (n) = 2n n is odd 2. Which of the functions in problem 1 are injective? Give a formal proof. 3. Prove or disprove and salvage: A × (B ∪ C) = (A × B) ∪ (A × C). 4. Let A ⊆ D and B ⊆ D. (a) Prove that f (A ∩ B) ⊆ f (A) ∩ f (B). (b) Prove: If f is injective, then f (A ∩ B) = f (A) ∩ f (B). 5. Let A ⊆ D. (a) Prove: If f is injective, then f (D \ A) ⊆ C \ f (A). (b) Prove: If f is surjective, then C \ f (A) ⊆ f (D \ A). 1