Question

Consider f,g:ℤ→ℤ given by:

• f(n) = 2n
• g(n) = n div 2

Which of the following statements are true? Select all that apply

(a)	f is an injection
(b)	g is an injection
(c)	f∘g is an injection
(d)	g∘f is an injection
(e)	f is a surjection
(f)	g is a surjection
(g)	f∘g is a surjection
(h)	g∘f is a surjection

Answer 1

So we have f(n)=2n and f:Z -> Z

but for 1 as output n must be 1/2 (not an integer) which is outside the domain, so it is not surjective but injective.

for g(n)= n div 2 and f: Z -> Z

and g is not even a function: g(1)=1/2 out of the range. (so neither surjective nor injective)

fog = 2 (n/2)= n hence is surjective.

gof = 2n / 2 = n again surjective.

so the right options are (a), (g) and (h).