A function, F: R × R ---› R x R has been defined as follows: F(x, y) = (3y - 1, 1 - x) for all (x, y) in R × R. Prove that F is a one-to-one correspondence that is, F is both One-to-one and Onto.

3. Let f A B be a function, and let Ao C A, BoC B. Prove that (a) f(f(f(Ao))) = f(Ao); (b) f(f(f¹(Bo))) = f-¹(Bo).

4) i) Let f: N→ C be a function with f(30) = 0, f(70) = 1 and f(105) = 1. Determine whether f is multiplicative or not. Explain why. Show that f(1) = 0 or f(1) = 1 if f is multiplicative. Determine whether the following functions are multiplicative or not. Give a detailed ii) iii) explanation. a) f(n)= ^ 1 if n 1 mod (2) 2 if n2 mod (2) b) f(n) = 1 if n 1 2 if n2 3 if n 0 mod (3) mod (3) mod (3)

Show that the function f : R → R given by f(x) = x3 is injective.