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October 11th, 2010, 07:24 AM   #1
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Equivalence Relation

Let S be the set of all ordered pairs (m,n) of positive integers. For (a1, a2) the belongs to S & (b1, b2) that belongs to S, define (a1, a2) ~ (b1, b2) if a1+b2 = a2+b1. Show that ~ is an equivalence relation.

I know that equivalence relation is possible is the following holds true: reflexivity, symmetry and transativity.

I am unsure of how to prove these though. Any help would be great.
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October 11th, 2010, 01:33 PM   #2
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Re: Equivalence Relation

reflexive:
Let (a1, a2) be in S. Then (a1, a2) ~ (a1, a2) simply means that
a1 + a2 = a2 + a1, which is true for integers.

symmetric : (a1, a2) ~ (b1, b2) => (b1, b2) ~ (a1, a2) ?
Assume that (a1, a2) ~ (b1, b2). This means that
a1 + b2 = a2 + b1, which is the same as
a2 + b1 = a1 + b2, which is the same as
b1 + a2 = b2 + a1, which is the definition of
(b1, b2) ~ (a1, a2)

transitive: Assume that (a1, a2) ~ (b1, b2) AND that (b1, b2) ~ (c1, c2). Show that (a1, a2) ~ (c1, c2)
Can you go from here?
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