1. Suppose f : S → S for some set S. Prove the following. (a) If f ° f is injective, then f is injective. (b) If f ° f is surjective, then f is surjective. 2. Given f : R → R and g : R → R, we define...

1. Suppose f : S → S for some set S. Prove the following.

(a) If f ° f is injective, then f is injective.

(b) If f ° f is surjective, then f is surjective.

2. Given f : R → R and g : R → R, we define the sum f + g by ( f + g)(x) = f (x) + g (x) and the product f g by ( f g)(x) = f (x) ⋅ g (x) for all x ∈
. Find counterexamples for the following.

(a) If f and g are bijective, then the sum f + g is bijective.

(b) If f and g are bijective, then the product f g is bijective.