1. ## Equivalence Relations

I'm having trouble comprehending the concept. For instance, I have this problem, asking whether or not this is an equivalence relation:

R = { (1,1), (1,2), (2,1), (2,2) , (3,3) }

on the set (1, 2, 3}

I understand that for this to be an equivalence relation it must be reflexive, symmetric, and transitive, but I am confused as to how to check for these properties.

For instance, I see reflexive properties in (1,1), (2,2), and (3,3) but not in the others. Is that a problem? And how do I check for transitive and symmetric?

2. As long as the set has, in this case, (1,1), (2,2), and (3,3), then its reflexive - the fact that 1 is ALSO, for example, equivalent to 2 has no bearing on the reflexivity. 1 is related to 1, 2 to 2, 3 to 3, and that's what you need.

For symmetry, you need to show that if a is related to b then b is related to a for ALL a and b.

Here, you don't need to worry about (1,1), (2,2), and (3,3) - clearly they're symmetric. Consider (1,2) - this means 1 is related to 2. For R to be symmetric, 2 must also be related to 1, i.e., (2,1) must be in your set (which it is). That takes care of everything - you've checked and "paired up" every element in R - note that (1,1), (2,2), and (3,3) are their own "pairs."

For transitivity, you need to show that if (a,b) is in R and (b,c) is in R, then (a,c) is in R. (In other words, if a is related to b and b is related to c, then a is related to c.)

Consider (1,2) in combination with (2,1) for example. We have:

1 is related to 2.
2 is related to 1.

The question for transitivity, then, is: "Is 1 related to 1"? It is, since (1,1) is also in the set.

Make sense?

Hope this helps!

3. Makes so much more sense than the book! Thank you very much!