Question

If $f:A\to B$ and $g:B\to C$ are one-one functions, show that gof is a one-one function.

Answer 1

f:A→Bf:A→B and g:B→Cg:B→C are both one-to-one functions.
Suppose a1,a2∈Aa1,a2∈A such that (gof)(a1)=(gof)(a2)(gof)(a1)=(gof)(a2)
⇒g(f(a1))=g(f(a2))⇒g(f(a1))=g(f(a2)) (definition of composition) Since gg is one-to-one, therefore,
f(a1)=f(a2)f(a1)=f(a2)
And since ff is one-to-one, therefore,
a1=a2a1=a2
Thus, we have shown that if (gof)(a1)=(gof)(a2)(gof)(a1)=(gof)(a2) then a1=a2a1=a2
Hence, gofgof is one-to-one function.