Question

Show that if f and g are one-to-one, then f o g is one-to one

Answer 1

We assume that:
f is 1-1
g is 1-1

Prove
f o g is 1-1.
Use definition of one to one ness

Answer 2

definition,
a function h(x) is one to one if and only
If h(x2) = h(x1) , then x2 = x1.

Answer 3

thats the contrapositive of
"distinct inputs have distinct outputs'

Answer 4

Proof:
Assume x1,x2 are two arbitrary elements in the domain of f o g
and that f(g(x2) = f(g(x1))

Since f is one to one f( g(x2) ) = f( g(x1) ) implies
g(x2)=g(x1).

Since g is one to one,
g(x2)=g(x1) implies x2 = x1.

We have shown that
f(g(x2) = f(g(x1)) implies x2 = x1. QED