Question

1. a) Let A = {2n|n ∈ ℤ} (ie, A is the set of even numbers) and define function f: ℝ → {0,1}, where f(x) = X_A(x) That is, f is the characteristic function of set A; it maps elements of the domain that are in set A (ie, those that are even integers) to 1 and all other elements of the domain to 0.

By demonstrating a counter-example, show that the function f is not injective (not one-to-one).

b) By demonstrating a counter-example, show that the function f: ℕ → ℕ, where f(x) = X² + 4, is not surjective (not onto).

c) Prove that function f : ℝ → ℝ, where f(x) = 5x+ 2, is injective (one-to-one).

d) (10 pts.)

Prove that function f: ℝ → ℤ, where f(x) = ⌊x− 4⌋, (note the use of the floor function) is surjective (onto).

2) Prove that function f:[0, ∞) → [0, ∞) where f(x) = 2x² , is a bijection ( (first, prove that it is oneto-one—injective; then prove that it is onto—surjective).