Let f : N → N be defined by State whether the function f is bijective. Justify your answer.

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Solution

The given function f:N→N is defined by,

f(n)={ n+1 2 , if n is odd n 2 , if n is even

Assume x 1 =5 and x 2 =6.

f( 5 )= 5+1 2 = 6 2 =3

f( 6 )= 6 2 =3

So, f( x 1 )=f( x 2 ) but x 1 ≠ x 2 .

Therefore, f is not one-one.

A function f:X→Y is onto or surjective, if for every y∈Y, there exists an element in X such that f( x )=y.

Let, n∈N be any element.

If n is odd, then n=2r+1 for some r∈N there exists 4r+1∈N such that,

f( 4r+1 )= 4r+1+1 2 =2r+1

If n is even, then n=2r for some r∈N there exists 4r∈N such that,

f( 4r )= 4r 2 =2r

So, f is onto.

Thus, f is onto but not one-one and thus fis not bijective.

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