Question

Show that function $f:N\to N$ given by $f\left(x\right)=3x$ is one-one but not onto.

Answer 1

Let $x_1,x_2ϵN$ be such that $f\left(x_1\right)=f\left(x_2\right)\Rightarrow 3x_1=3x_2\Rightarrow x_1=x_2$

Thus, $f\left(x_1\right)=f\left(x_2\right)\Rightarrow x_1=x_2$ then f is one-one

As $1ϵN$ and there does not exist any $xϵN$ such that $f\left(x\right)=1$

so, f is not onto

Hence proved.