Relations Functions

A function f from the set of natural numbers to integers defined by

f(n) = (n-1)/2, when n is odd

f(n) = -n/2, when n is even

  1. one-­one and onto both
  2. one-­one and but not onto
  3. neither one-­one nor onto
  4. onto but not one-­one

Solution

If n is odd, let n = 2k - 1

f(2k1 - 1) = f(2k2 - 1)

2k1 - 1 + 1/2 = 2k2 - 1 + 1/2

k1 = k2

f(n) is one-one function, if n is odd.

If n is even, let n = 2k

f(2k1) = f(2k2)

k1 = k2

f(n) is one-one function, if n is even.

Also, f'(n) is increasing, when n is odd and f'(n) is decreasing, when n is even.

The correct option is A.