   Chapter 2.1, Problem 23E

Chapter
Section
Textbook Problem

In Exercises 13 − 24 , prove the statements concerning the relation < on the set Z of all integers. z − x < z − y if and only if y < x .

To determine

To prove: That zx<zy holds if and only if y<x on the set of all integers.

Explanation

Formula Used:

For each xZ, there is an additive inverse of x in Z, denoted by x, such that x+(x)=0=(x)+x.

Proof:

For all x,y,zZ, zx<zy means (zy)(zx)>0 or (zy)(zx)Z+.

Then,

(zy)(zx)=z+(y)+(z)+x

Rearrange the terms by commutative law for addition

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