Show that f : N → N, given by f(x)={ x+1, if x is odd, and x-1, if x is even} is both one-one and onto.

Class 12, NCERT Chapter 1,  Example12

Suppose f(x₁ ) = f(x₂ ).
Note that if x₁ is odd and x₂ is even, then we will have
x₁ + 1 = x₂ – 1, i.e., x₂ – x₁ = 2, which is impossible.
Similarly, the possibility of x₁ being even and x₂ being odd can also be ruled out, using a similar argument.
Therefore, both x₁ and x₂ must be either odd or even.
Suppose both x₁ and x₂ are odd.
Then f(x₁ ) = f(x₂ )
⇒ x₁ + 1 = x₂ + 1
⇒ x₁ = x₂
Similarly, if both x₁ and x₂ are even, then also
f(x₁ ) = f(x₂ )
⇒ x₁ – 1 = x₂ – 1
⇒ x₁ = x₂ .
Thus, f is one-one.
Also, any odd number 2r + 1 in the co-domain N is the image of 2r + 2 in the domain N and
any even number 2r in the co-domain N is the image of 2r – 1 in the domain N. 

Thus, f is onto.