6. Prove or disprove the statements: (a) The function f: Z × Z → Z defined by f (m,n) = m + n + 1 is onto. b) The function f:Z × Z → Z defined by f(m,n) = m + n + 1 is one-to-one.

1. Define the function f : Z x Z → Z by 2. Suppose a, b are natural numbers with gcd(a, b) f(m, n) = ma + nb. (a) Show that f is not one-to-one. (b) Show that f is onto.

2 (a) Prove that, if a and b are nonnegative integers such that a 4 b, then 3ª = 36. (b) Using (a), explain why the below defines a well-defined function f: ƒ: Z4 → Zs f(a)) = [3]s if a ≥ 0.

9. Determine whether f is injective, surjective or bijective. a. Suppose f: N→ N has the rule f(n) = 4n+ 1. b. Suppose f: Z → Z has the rule f(n) = 3n² – 1. c. Suppose f:Z-Z has the rule f(n)-3n-1. d. Suppose f : N → N has the rule f(n) – 4n² + 1. e. Suppose f: R → R where f(x) = [x/2].