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On the set N x N, define the following relation:

(a, b) ~ (c, d) if and only if a + d = b + c

(a). Show that this is an equivalence relation

I have shown that this is an equivalence relation by proving that the relation has a reflexive, symmetric, and transitive property.

(b). Describe the equivalence class of (1, 1)

I have a rather crude understanding of describing an equivalence class, but as far as I know, the equivalence class of (1, 1) is any value for (c, d) ∈ N x N in which 1 + d = 1 + c. So far, my guess is to describe (c, d) such that c = d, c - d = 0, 2 d = c + d = 2 c. Is there a 'Proper' way to describe it?

amundi12
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    You can say it's the subset consisting of (c,d) such that c=d, or you can say it's all elements of the form (c,c) where c is in $N$. It's the same. – Not a grad student Dec 10 '14 at 23:21
  • Do you have any suggestions to how many descriptions I should give? I mean, there are lots of ways to describe it. c = d, c - d = 0, 2d = c + d = 2c and thats just to name a few – amundi12 Dec 10 '14 at 23:22
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    Since they're all the same, you should just give one. But c=d is the simplest so I would go with that one. Or (c,c) where c is in N is perhaps simpler. – Not a grad student Dec 10 '14 at 23:24

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