Reference no: EM13334443

1. Consider functions f : X → Y and g : Y → Z.
(a) Suppose that f is surjective. Is g ? f surjective? Either prove it is, or give a counterexample.
(b) Suppose that g is surjective. Is g ? f surjective? Either prove it is, or give a
counterexample.
(c) Suppose that both f and g are surjective. Is g ? f surjective? Either prove it is, or give a counterexample.
(d) Suppose that g ? f is surjective. Is g surjective? Either prove it is, or give a counterexample.
(e) Suppose that g ? f is surjective. Is f surjective? Either prove it is, or give a counterexample.

2. Let f : X → Y be a function.
(a) Let A and B be subsets of X. Define the set A B.
(b) Prove that
f[A] f[B] ⊆ f[A B].
(c) Now suppose that f : X → Y is injective. Prove that
f[A B] ⊆ f[A] f[B].
(d) Give an example of a function f : X → Y together with sets A, B ⊆ X such that
f[A B] 6= f[A] f[B].