   Chapter 2.1, Problem 14E

Chapter
Section
Textbook Problem

In Exercises 13 − 24 , prove the statements concerning the relation < on the set Z of all integers.If x < y and z < w , then x + z < y + w .

To determine

To prove: The statement, ‘If x<y and z<w, then x+z<y+w’ on the set of integers.

Explanation

Formula Used:

Subtraction:

Subtraction is defined by xy=x+(y) for arbitrary x and y in Z.

The order relation less than:

For integers x and y, x<yifandonlyifyxZ+,

Where yx=y+(x).

x+(y+z)=(x+y)+z for arbitrary elements x,y,zZ.

The distributive law:

x(y+z)=xy+xz holds for all elements x,y,zZ.

x+y=y+x for arbitrary x,yZ.

Z+, the set of positive integers is closed under addition:

If x,yZ+, then x+yZ+.

Proof:

Consider the given information.

x<yyxZ+bydefinitionoforderrelationlessthanAnd,z<wwzZ+bydefinitionoforderrelationlessthan

Consider x,y,z,wZ

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