equivalence

**mcfc** · Nov6-09, 05:26 PM

If I have a subset, how do I define an equivalence relation.
I understand it has to satisfy three properties:transitive, symmetric and reflexive, but I'm not sure how to give an explicit definition of the equivalence relation, for example on I where
$LaTeX Code: I=\\{(x,y) : 0 \\le x\\le 1 \\ \\& \\ 0 \\le y \\le 1\\}$

**CharmedQuark** · Nov6-09, 09:11 PM

Do you know what a Cartesian product is? If you don't its a very important topic for anyone learning set theory to know.

If you do, then an equivalence relation R from A to B is a subset of A X B. In other words an equivalence relation R contains those ordered pairs (a,b) $LaTeX Code: \\in$ A X B such that a is related to b by R.

In your example that equivalence relation is a subset of $LaTeX Code: \\Re$ X $LaTeX Code: \\Re$ consisting of those (x,y) $LaTeX Code: \\in$ $LaTeX Code: \\Re$ X $LaTeX Code: \\Re$ such that 0 $LaTeX Code: \\leq$ x $LaTeX Code: \\leq$ 1, 0 $LaTeX Code: \\leq$ y $LaTeX Code: \\leq$ 1.

Hope that makes sense to you.

**mcfc** · Nov7-09, 11:55 AM

Originally Posted by CharmedQuark

Do you know what a Cartesian product is? If you don't its a very important topic for anyone learning set theory to know.

If you do, then an equivalence relation R from A to B is a subset of A X B. In other words an equivalence relation R contains those ordered pairs (a,b) $LaTeX Code: \\in$ A X B such that a is related to b by R.

In your example that equivalence relation is a subset of $LaTeX Code: \\Re$ X $LaTeX Code: \\Re$ consisting of those (x,y) $LaTeX Code: \\in$ $LaTeX Code: \\Re$ X $LaTeX Code: \\Re$ such that 0 $LaTeX Code: \\leq$ x $LaTeX Code: \\leq$ 1, 0 $LaTeX Code: \\leq$ y $LaTeX Code: \\leq$ 1.

Hope that makes sense to you.

HI

That does makes sense, but I can't see how to define an explicit equivalence relation...?

**bpet** · Nov7-09, 08:00 PM

Originally Posted by mcfc

...for example on I where
$LaTeX Code: I=\\{(x,y) : 0 \\le x\\le 1 \\ \\& \\ 0 \\le y \\le 1\\}$

I x I has the required properties, right?

**mcfc** · Nov7-09, 09:00 PM

Originally Posted by bpet

I x I has the required properties, right?

sorry...I don't follow(again)

**CharmedQuark** · Nov9-09, 10:05 AM

Originally Posted by mcfc

sorry...I don't follow(again)

The equivalence relation you gave is a relation on the set I. I X I is the Cartesian product of I with itself. Since the relation R is from I to I it is a subset of I X I. An equivalence relation is a set and can be written as such.

Perhaps if you rephrased your question I could be of more help?