How Do I Define an Equivalence Relation on a Subset?

[mcfc]

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
$I=\{(x,y) : 0 \le x\le 1 \ \& \ 0 \le y \le 1\}$

 

[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) $$\in$$ A X B such that a is related to b by R.

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

Hope that makes sense to you.

 

[mcfc]

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) $$\in$$ A X B such that a is related to b by R.

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

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

I x I has the required properties, right?

 

[mcfc]

I x I has the required properties, right?

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

 

[CharmedQuark]

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?