Math Is Fun Forum

Kazy

Member

Registered: 2006-01-24

Posts: 37

Equivalence Relation

I need to prove that R is an equivalence relation S in the following:

S = {(a,b) ∈ Z x Z | b ≠ 0}. R is the relatoin on S defined by (a,b)R(c,d) if ad=bc.

Can anyone help? I'm completely lost.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Equivalence Relation

Just take it one step at a time.

We must first show tha a~a.

so is (a, b) ~ (a, b)?

That is, ab = ba?  If you are doing multiplication (which I assume), then yes, that statement is true.

Now let's assume a ~ b.  Show that b ~ a.

Since a ~ b, ad = bc.  So cb = da, and thus, b ~ a.

Now you try transitivity.  Assume that a ~ b and b ~ c.  Try to show that a ~ c.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."