(a) Someone has claimed that since 14 is itself a multiple of 7, then there are more multiples of 7 than there are of 14. Show this is false by establishing 7Z - 14Z. (Do this formally, by establishing the appropriate function.)

Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section1.1: Real Numbers
Problem 35E
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Q3
(Recall the notation: For any set S CR and any k e R, kS = {ks : s E S}.)
(a) Someone has claimed that since 14 is itself a multiple of 7, then there are more
multiples of 7 than there are of 14. Show this is false by establishing 7Z - 14Z. (Do
this formally, by establishing the appropriate function.)
(b) This person went on further to claim that there are more integers that are not
divisible by 7 than are divisible by 7. (So, they are claiming that the cardinality of the
set of integers not divisible by 7 is larger than the cardinality of the integers divisible by
7.)
Is this claim true? (Justify your answer, but the justification can be informal.)
Transcribed Image Text:Q3 (Recall the notation: For any set S CR and any k e R, kS = {ks : s E S}.) (a) Someone has claimed that since 14 is itself a multiple of 7, then there are more multiples of 7 than there are of 14. Show this is false by establishing 7Z - 14Z. (Do this formally, by establishing the appropriate function.) (b) This person went on further to claim that there are more integers that are not divisible by 7 than are divisible by 7. (So, they are claiming that the cardinality of the set of integers not divisible by 7 is larger than the cardinality of the integers divisible by 7.) Is this claim true? (Justify your answer, but the justification can be informal.)
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