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- Show that f:N=N given be f(x)={x+1, if x is odd
- x-1, if x is even } is bijectve .

Let *m, n N*.

If *n* and *m* are odd, then,

If *n* and *m* are even, then,

Thus, we have *f(n) = f(m) ⇒ n = m* in both the cases.

If n is odd and m is even, then* f(n) = n + 1* is even and *f(m) = m - 1* is odd. Therefore, *n m ⇒ f(n) f(m). *Similarly, if *n* is even and *m* is odd, then, *n m ⇒ f(n) f(m).*

Hence, *f* is **injective**.

Let* n N*.

If *n* is an odd natural number, there exists an even number *n - 1 N, *such that

*f (n - 1) = n - 1 + 1 = n*

If *n* is an even natural number, there exists an odd number *n + 1 N, *such that

*f (n + 1) = n + 1 - 1 = n*

Also, *f(1) = 0*.

Thus, every element of *N* has its pre-image in *N*. So, *f* is **surjective** (onto function).

Hence, *f* is **bijective**.

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