Let *f*: **N** → **N** be defined by f(n) = `{((n+1)/2, "if n is odd"),(," for all n ∈ N"), (n/2, if "n is even"):}`

State whether the function f is bijective. Justify your answer.

Advertisement Remove all ads

#### Solution

*f*: **N** → **N** is defined as f(n) = `{((n+1)/2, "if n is odd"),(," for all n ∈ N"), (n/2, if "n is even"):}`

It can be observed that:

`f(1) = (1+1)/2 = 1` and `f(2) = 2/2 = 1` [By definition of f]

`:. f(1) = f(2), "where " 1 != 2`

∴ *f* is not one-one.

Consider a natural number (*n)* in co-domain **N**.

Case **I: ***n* is odd

∴*n* = 2*r* + 1 for some *r* ∈ **N. **Then, there exists 4*r *+ 1∈**N** such that

`f(4r + 1) = (4r + 1 + 1)/2 = 2r + 1`

Case **II: ***n* is even

∴*n* = 2*r* for some *r* ∈ **N. **Then,there exists 4*r* ∈**N** such that `f(4r) = (4r)/2 = 2r`

∴ *f* is onto.

Hence, *f* is not a bijective function.

Concept: Types of Functions

Is there an error in this question or solution?

Advertisement Remove all ads

#### APPEARS IN

Advertisement Remove all ads

Advertisement Remove all ads