# Let f : N →

TIP: One – One Function: – A function is said to be a one – one functions or an injection if different elements of A have different images in B.

So, is One – One function

a≠b

f(a)≠f(b) for all f(a) = f(b)

a = b for all Onto Function: – A function is said to be a onto function or surjection if every element of A i.e, if f(A) = B or range of f is the co – domain of f.

So, is Surjection iff for each , there exists such that f(a) = b

Bijection Function: – A function is said to be a bijection function if it is one – one as well as onto function.

Now, suppose

f(n1) = f(n2)

If n1 is odd and n2 is even, then we have

n1 + 1 = n2 – 2

n2 – n1 = 2

Not possible

Suppose both n1 even and n2 is odd.

Then, f(n1) = f(n2)

n1 – 1 = n2 + 1

n1 – n2 = 2

Not possible

Therefore, both n1 and n2 must be either odd or even

Suppose both n1 and n2 are odd.

Then, f(n1) = f(n2)

n1 + 1 = n2 + 1

n1 = n2

Suppose both n1 and n2 are even.

Then, f(n1) = f(n2)

n1 – 1 = n2 – 1

n1 = n2

Then, f is One – One

Also, any odd number 2r + 1 in the co – domain N will have an even number as image in domain N which is

f(n) = 2r + 1

n – 1 = 2r + 1

n = 2r + 2

Any even number 2r in the co – domain N will have an odd number as image in domain N which is

f(n) = 2r

n + 1 = 2r

n = 2r – 1

Thus f is Onto function.

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