2 10 pts Let f A B and g B C be functions and consider the

2 (10 pts). Let f : A B and g : B C be functions, and consider the composition h = g f : A C.

(a). Show that if f and g are both injective, then h is injective.

(b). Show that if h is injective, then f is injective. Give an example showing that g does not have to be injective

Solution

a)

Let, h(a)=h(b)

So,

g(f(a))=g(f(b))

g is injective so f(a)=f(b)

f is injective so

a=b

Hence, h is injective

b)

Let, f(a)=f(b)

Then

g(f(a))=g(f(b))

ie h(a)=h(b)

But since h is injective

a=b

Hence, f is injective

Let, f(x)=x, A=R+ ie set of positive real numbers

B=R, C=R

f is then injective,

g(x)=x^2 , g is from R to R so g is not injective

But, f(g(x))=x^2 is from R+ to R so f(g(x))=h(x) is injective