Question

In: Advanced Math

Determine whether or not the following is an equivalence relation on N. If it is not,...

Determine whether or not the following is an equivalence relation on N. If it is not, determine which property fails. If it is, prove it.

(a) x ∼ y if and only if x|y (x evenly divides y).

(b) x ∼ y if and only if the least prime dividing x is the same as the least prime dividing y. In this case, instead if using N, use the set A = {2,3,4,5,...}.

(c) x∼yifandonlyif|x−y|<5.

(d) x∼yifandonlyifx=4.

2. Prove that x ∼ y if and only if sin x = sin y is an equivalence relation on R. What is the equivalence class of [π]?

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