Question

Need help proving: For every nonempty set S, there is precisely one antisymmetric equivalence relation on S

Answer 1

I know what an equivalence relation is (and its 3 axioms: symmetry, transitivity and reflexivity). But what is an antisymmetric equivalence relation?

Answer 2

If R is such a relation, then .... what?

Answer 3

It's an equivalence relation but with the added thing that [sRt ^ tRs] iff s=t...so basically you have the most primitive of equivalence relations (x,x)

Answer 4

ok.

Answer 5

just one variable, and it satisfies reflexivity, symmetry and transitivity

Answer 6

james; please have a look:
http://openstudy.com/groups/mathematics#/groups/mathematics/updates/4e9635690b8bedaecf3bb195

Answer 7

It's ok I think I've solved mine, I will suppose that another one exists and show that that they are equal

Answer 8

Well there's a trivial antisymmetric relation: xRy if and only if x = y.
And I now I suppose we need to show that if any other relation is antisymmetric it is this trivial relation.

Answer 9

So suppose T is an arbitrary antisymmetric relation on S.
Then we want to show somehow that xTy implies x = y. Then T = S.

So to do that we want to show somehow that xTy implies yTx and then we can use the antisymmetry property of T.

I'll need to think a bit more about how to do this; I'm experimenting with the reflexivity and transitivity properties but don't have it right now.

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Anyway, assuming it is true, it's now obvious why antisymmetric relations are useless!

Answer 10

yeah, well they are from what I gather the smalles of equivalence relations