Question

Explain how a proof by cases can be used to prove a result about absolute values, such as the fact that\(\left| {xy} \right| = \left| x \right|\left| y \right|\) for all real numbers\(x and y\).

Short answer

\(\left| {xy} \right| = \left| x \right|\left| y \right|\)is true for all the cases of values of \(x and y\).

Step-by-step solution

Introduction

In the proof of \(\left| {xy} \right| = \left| x \right|\left| y \right|\), for all real numbers\(x and y\)it will need four cases. If any of\(x and y\) then the result is trivial.

Note that in proof remove absolute values and use the fact that\(\left| a \right| = a\)when \(a \ge 0\)and \(\left| a \right| = - a\)when\(a < 0\).

Case 1. \(x and y\)both are positive

This implies\(x. y\)will be positive.

So,\(\left| {xy} \right| = xy\)

Now, \(\left| x \right| = x\)

And\(\left| y \right| = y\)

Hence, \(\left| {xy} \right| = \left| x \right|\left| y \right|\)

Case 2. x is positive and y is negative

Now, y is negative implies there exist a positive value c such that \(y = - c\)

Hence,

\(\begin{array}{l}x. y = x.\left( { - c} \right)\\\,\,\,\,\,\,\,\,\,\, = - xc\end{array}\)

Now, \(\left| x \right| = x\)

And, \(\begin{array}{l}\left| y \right| = \left| { - c} \right|\\\,\,\,\,\,\, = c\end{array}\)

Hence, \(\left| {xy} \right| = \left| x \right|\left| y \right|\)

Case 3. x is negative and y is positive

Now, x is negative implies there exist a positive number c such that \(x = - c\)

Hence,

\(\begin{array}{l}x. y = \left( { - c} \right).y\\\,\,\,\,\,\,\,\,\,\, = - cy\end{array}\)

This implies \(x. y\)is negative.

Now take absolute value and get;

\(\begin{array}{}\left( {x. y} \right) = \left| {\left( { - c} \right).y} \right|\\\,\,\,\,\,\,\,\,\,\,\,\,\,\, = cy\end{array}\)

Now, \(\begin{array}{}\left| x \right| = \left| { - c} \right|\\\,\,\,\,\,\, = c\end{array}\)

And, \(\left| y \right| = y\)

Hence,\(\left| {xy} \right| = \left| x \right|\left| y \right|\)

Case 4. \(x and y\)both are negative

Now, \(x and y\)both are negative implies there exist positive numbers c and d such that\(x = - c\)and\(y = - d\)

Hence,

\(\begin{array}{}x. y = \left( { - c} \right).\left( { - d} \right)\\\,\,\,\,\,\,\,\,\,\, = cd\\ \Rightarrow \left| {xy} \right| = cd\end{array}\)

Now,

\(\begin{array}{}\left| x \right| = \left| { - c} \right|\\\,\,\,\,\,\, = c\end{array}\)

And,

\(\begin{array}{}\left| y \right| = \left| { - d} \right|\\\,\,\,\,\,\, = d\end{array}\)

Hence, \(\left| {xy} \right| = \left| x \right|\left| y \right|\)

Thus proved.