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.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Let Q: ℤ⁺−{1} → ℙ(ℤ⁺) be a function defined as Q(n) = {p₁,...,p_k}, where p₁,...,p_k are the prime factors of n. That is, n = p₁^e1⋅ p_k^ek, for some positive integers e₁,...,e_k.

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

Show that R_Q is an equivalence relation.

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