   Chapter 1.4, Problem 9E

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Textbook Problem
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# The definition of an even integer was stated in Section 1.2. Prove or disprove that the set E of all even integers is closed with respect toa. addition defined on Z .b. multiplication defined on Z .

(a)

To determine

Whether the set E of all even integers is closed with respect to addition defined on Z where E={...,6,4,2,0,2,4,6,...}.

Explanation

Consider the set of all even integer E.

Even integers are of the form 2k for some integer k.

The addition operation on Z is defined as

xy=x+y, where x,yZ

For some integers k and m substitute 2k for x and 2m for y in xy=x+y

(b)

To determine

Whether the set E of all even integers is closed with respect to multiplication defined on Z where E={...,6,4,2,0,2,4,6,...}.

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