Let $A=\{-1,0,1\}$ and $f=\left\{\left(x, x^{2}\right): x \in A\right\}$. Show that $f: A \rightarrow A$ is neither one-one nor onto.

Solution:

$A=\{-1,0,1\}$ and $f=\left\{\left(x, x^{2}\right): x \in A\right\}$ Given, $f(x)=x^{2}$ Given, $f(x)=x^{2}$

Injectivity:
$f(1)=1^{2}=1$ and
$f(-1)=(-1)^{2}=1$

$\Rightarrow 1$ and $-1$ have the same images.
So, $f$ is not one-one.

Surjectivity:
Co-domain of $f=\{-1,0,1\}$
$f(1)=1^{2}=1$
$f(-1)=(-1)^{2}=1$ and
$f(0)=0$
$\Rightarrow$ Range of $f=\{0,1\}$
So, both are not same.
Hence, $f$ is not onto.