Question

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

Answer 1

Given, f : A → B and g : B → C are one - one.
Then, gof : A → B
Let us take two elements x and y from A, such that
$$\begin{gathered} \left(gof\right)\left(x\right)=\left(gof\right)\left(y\right) \\ \Rightarrow g\left(f\left(x\right)\right)=g\left(f\left(y\right)\right) \\ \Rightarrow f\left(x\right)=f\left(y\right)\left(\text{As,}g\text{is one-one}\right) \\ \Rightarrow x=y\left(\text{As,}f\text{is one-one}\right) \end{gathered}$$
Hence, gof is one-one.