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October 11th, 2010, 07:24 AM  #1 
Newbie Joined: Sep 2010 Posts: 10 Thanks: 0  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. 
October 11th, 2010, 01:33 PM  #2 
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4  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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