Give two examples of functions from Z to Z that one-to-one but not onto.

To find:

The two examples of function from ℤ to ℤ that are one-to-one but not onto.

Given information:

One-to-one function but not onto.

Concept used:

Rational number can be written in the form of pq, where p,q∈ℤ and q≠0.

Calculation:

Example 1:

Define a function f:ℤ→ℤ such that f(n)=2n

Then f is one-to-one but not onto.

Take

f(n1)=f(n2)⇒2n1=2n2⇒n1=n2

Therefore, f is one-to-one function.

Now ,

Let s=f(n) then

n=f−1(s)s=2nn=s2

Then

f−1(s)=s2f−1(n)=n2∉ℤ

Therefore, f is not onto