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.