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Let Q: ℤ+−{1} → ℙ(ℤ+) be a function defined as Q(n) = {p1,...,pk}, where p1,...,pk are the prime factors of n. That is, n = p1e1 ⋅ pkek, for some positive integers e1,...,ek. Let RQ be a relation on ℤ+−{1} such that, (x, y) ∈ RQ if and only if Q(x) = Q(y). Show that RQ is an equivalence relation.

Let Q: ℤ+−{1} → ℙ(ℤ+) be a function defined as Q(n) = {p1,...,pk}, where p1,...,pk are the prime factors of n. That is, n = p1e1 ⋅ pkek, for some positive integers e1,...,ek. Let RQ be a relation on ℤ+−{1} such that, (x, y) ∈ RQ if and only if Q(x) = Q(y). Show that RQ is an equivalence relation.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.2: Mappings
Problem 19E
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Let Q: ℤ+−{1} → ℙ(ℤ+) be a function defined as Q(n) = {p1,...,pk}, where p1,...,pk are the prime factors of n. That is, n = p1e1 ⋅ pkek, for some positive integers e1,...,ek.

Let RQ be a relation on ℤ+−{1} such that, (x, y) ∈ RQ if and only if Q(x) = Q(y).

Show that RQ is an equivalence relation.

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