1. Let S = {1, 2, 3, 4, 5} and let f, g, h : S → S be the functions defined by f = {(1, 2),(2, 1),(3, 4),(4, 5),(5, 3)} g = {(1, 3),(2, 5),(3, 1),(4, 2),(5, 4)} h = {(1, 2),(2, 2),(3, 4),(4, 3),(5, 1)}.
(a) Explain why f and g have inverses but h does not.
(b) Show that (f ? g) -1 = g -1 ? f -1 but (f ? g) -1 6= f -1 ? g -1 .
2. Let A = {x ∈ R | x 6= -3} and define f : A → R by f(x) = x - 3 x + 3.
(a) Show that f is one-to-one.
(b) Find rng(f).
(c) Define g : A → rng(f) by g(x) = f(x) for all x ∈ A. Find g -1 .
3. Show that the given sets have the same cardinality by finding a bijection between them. In each part, state whether the pair of sets is finite, countably infinite, or uncountable. (a) {Z, Q, R} and {1, 2, 3}. (b) Z and 2Z + 1. (c) The intervals (2, 4) and (1, 7). (d) (0, π 2 ) and R +. (e) Z + and a 2 b | a, b ∈ Z + .
4. (a) Without giving a bijection, explain why the set of all prime numbers is countable.
(b) Explain why the set of all continuous functions f : R → R is uncountable.
5. Suppose A and B are sets with A $ B. If A and B are infinite, is it necessarily true that |A| < |B|? Give a proof or a counterexample.
6. Suppose A and B are countably infinite sets. Show that A × B is countably infinite by proving that |A × B| = |Z + × Z +|.