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Let Q be the set {(a, b) ∶ a ∈ Z and b ∈ N}. Let...

Let Q be the set {(a, b) ∶ a ∈ Z and b ∈ N}.

Let (a, b), (c, d) ∈ Q. Show that (a, b) ∼ (c, d) if and only if ad − bc = 0 defines an equivalence relation on Q.

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Answer #1

​​​​​​a relation is an equivalence relation if it is

1)reflexive:a relation is reflexive if (a,b) related to (a,b) on Q

2)symmetric:a relationis symmetric if (a,b) related to (c,d) then (c,d) related to (a,b)

3) transitive:a relation is transitive if (a,b) related to (c,d) and (c,d) related to (e,f) then (a,b) related to (e,f).

Here (a,b),(c,d),(e,f) are elements on Q.

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