f: R → R :f(x)= {1, if x is rational -1, if x is irrational show that f is many one and into

Question

\(f:R→R :f(x)=\begin{cases}1,\text{ if x is rational}\\ -1,\text{ if x is irrational}\end{cases}\)

Show that f is many-one and into.

Answer 1

To prove: function is many-one and into

Given: \(f:R→R :f(x)=\begin{cases}1,\text{ if x is rational}\\ -1,\text{ if x is irrational}\end{cases}\)

We have,

f(x) = 1 when x is rational

It means that all rational numbers will have same image i.e. 1

⇒ f(2) = 1 = f (3) , As 2 and 3 are rational numbers

Therefore f(x) is many-one

The range of function is [{-1},{1}] but codomain is set of real numbers.

Therefore f(x) is into