Chapter 3, Problem 1RE

Define $f:\text{Z×Z}\to \text{Z}$ by $f\left(a,b\right)=ab$. Determine whether f is one-to-one.

To determine

Whether f is one-to-one or not.

Answer to Problem 1RE

Solution: f is not one-to-one.

Explanation of Solution

Given Information:

Define $f:\text{Z×Z}\to \text{Z}$ by $f\left(a,b\right)=ab$

Explanation:

For one-one

$\left(a,b\right)=\left(c,d\right)$ if $f\left(a,b\right)=f\left(c,d\right)$.

As,

$\begin{array}{l}f\left(a,b\right)=f\left(c,d\right)\\ ab=cd\end{array}$

But a and b are not always equal to c and d to satisfy the above condition.

For example:

$\begin{array}{l}f\left(2,2\right)=4=f\left(3,1\right)\\ \left(2,2\right)\ne \left(3,1\right)\end{array}$