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  • Explain solution RD Sharma class 12 chapter Functions exercise 2.1 question 23 maths

Explain solution RD Sharma class 12 chapter Functions exercise 2.1 question 23 maths

Answers (1)

Given:

     f(n)=\left\{\begin{array}{ll} n+1, & \text { if } n \text { is odd } \\ n-1, & \text { if } n \text { is even } \end{array}\right.       

To prove:

Function f is bijection.

Hint:

For any function to be a bijection, the given function should be one-one and onto.

Solution:

Here f:N \rightarrow N

First we will check whether the given function is one-one or not.

For one-one,

Case I : If n is odd

Let x,y\in N such that f(x)=f(y)

As        f(x)=f(y)

\begin{array}{ll} \Rightarrow & x+1=y+1 \\ \Rightarrow & x=y \end{array}

Case II : If n  is even

Let x,y\in N such that f(x)=f(y)

As        f(x)=f(y)

\begin{array}{ll} \Rightarrow & x-1=y-1 \\ \Rightarrow & x=y \end{array}

So, f is injective (one-one).

Again we will check whether the given function is onto or not.

For onto,

Case I : If n  is odd

As for everyf:N \rightarrow N,n \in N , there exists y=n-1    

f(y )=f(n-1)                                                                     

=n-1+1

  =n

Case II : If n is even

As for every n \in N, there existsy=n+1

\Rightarrow f(y)=f(n+1)=n+1-1=n

So f is surjective (onto).

Hence f is a bijection.

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