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).
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