[14 marks] Working with functions. In this question, we will explore various properties of functions.

You may want to review the basic definitions and terminology introduced on pages 15-16 of the course

notes. Then, read the following definitions carefully.

Definition: A function f : A B is one-to-one iff no two elements of A have the same image. Symbolically,

a1, a2 A, f(a1) = f(a2) a1 = a2. (3)

Definition: A function f : A B is onto iff every element of B is the image of at least one element

from A. Symbolically,

b B, a A, f(a) = b. (4)

Definition: For all functions f : A B and g : B C, their composition is the function g f : A C

defined by:

a A, (g f)(a) = g(f(a)). (5)

(a) [4 marks]

i. Prove that g1 : Z Z; g1(x) = x 4 is both one-to-one and onto.

ii. Prove that g2 : R R; g2(x) = |x| + x is neither one-to-one nor onto.

(b) [4 marks] Give explicit, concrete definitions for two functions f1, f2 : Z Z+ such that:

i. f2 is onto but not one-to-one,

ii. f1 is one-to-one but not onto,

and prove that each of your functions has the desired properties.

(c) [6 marks] Let f : A B and g : B C be arbitrary functions. Prove or disprove each of the

following. In each case, first write down in symbolic notation the exact statement you are attempting

to prove (either the original statement or its negation).

i. If g f is one-to-one, then f is also one-to-one.

ii. If g f is onto, then g is also onto.

iii. If g f is both one-to-one and onto, then f and g are also both one-to-one and onto.