   Chapter 2.1, Problem 29E

Chapter
Section
Textbook Problem

For an integer x , the absolute value of x is denoted by | x | and is defined by | x | = { x                 if   0 ≤ x − x             if   x<0 Use this definition for the proofs in Exercises 28 − 30 .Prove that | x y | = | x | ⋅ | y | for all x and y in Z .

To determine

To prove: The expression |xy|=|x||y| for all x and y in Z.

Explanation

Formula Used:

For any integer x the absolute value is denoted by |x| defined as

|x|={xifx0xifx<0

Proof:

Consider the four cases that arises for the expression |xy|=|x||y|.

Case I: When x0,y0

If x=0, then xy=0y=0. So, |xy|=0. Moreover, |x|=|0|=0, so |xy|=|x||y|.

If y=0, then xy=x0=0. So, |xy|=0. Moreover, |y|=|0|=0, so |xy|=|x||y|.

If x0 and y0, then x,yZ+ and xyZ+ as set of positive integers is closed with respect to multiplication.

Now we have 0<xy, so, |xy|=xy. On the other hand, 0<x and 0<y means that xy=|x||y|.

So, |xy|=|x||y|

Case II: When x0,y<0

If x=0, then we conclude that |xy|=0=|x||y|.

If x0, then xZ+ and yZ, so yZ+. Thus, x(y)=(xy)Z+ as Z+ is closed with respect to multiplication. This means that xyZ, so xy<0.

Now we know that |xy|=(xy), |x|=x and |y|=y

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Use an integral to estimate the sum i=110000i.

Single Variable Calculus: Early Transcendentals, Volume I 