For a natural number n > 1, let A (n) be the set of all primes that divide n. For example: 

 

A (15) = {3, 5} since 15 = 3 ∙ 5 

A (14000) = {2, 5, 7} since 14000 = 24 ∙ 53 ∙ 7 

A (5) = {5} 

 

Let T ⊆ ℕ × ℕ be a relation (a binary relation) on the set of natural numbers defined as follows:

 

(x, y) ∈ R ⟺ A(x) ⊆ A(y). 

 

a) Is R reflexive? Prove your answer. 

b) Is R symmetric? Prove your answer. 

c) Is R antisymmetric? Prove your answer. 

d) Is R transitive? Prove your answer. 

e) Is R an equivalence relation? Prove your answer. 

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