$f: R \rightarrow R: f(x)=\left\{\begin{array}{c} 1, \text { if } x \text { is rational } \\ -1, \text { if } x \text { is rational } \end{array}\right.$ Show that $\mathrm{f}$ is many-one and into.

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$f: R \rightarrow R: f(x)=\left\{\begin{array}{c} 1, \text { if } x \text { is rational } \\ -1, \text { if } x \text { is rational } \end{array}\right.$ Show that $\mathrm{f}$ is many-one and into.

Solution:

We need to prove: function is many-one and into
It is given that: $f: R \rightarrow R: f(x)=\left\{\begin{array}{c}1, \text { if } x \text { is rational } \\ -1, \text { if } x \text { is irrational }\end{array}\right.$
We have,
$f(x)=1$ when $x$ is rational
So, all rational numbers will have same image i.e. 1
$\Rightarrow f(2)=1=f(3)$ , As 2 and 3 are rational numbers
So, $f(x)$ is many-one
The range of function is $[\{-1\},\{1\}]$ but codomain is set of real numbers.
Thus $f(x)$ is into