# Equivalence Relations

1. Jun 17, 2012

### hammonjj

1. The problem statement, all variables and given/known data
Which of these relations on {0, 1, 2, 3} are equivalence relations? Determine the properties of an equivalence relation that the others lack

a) { (0,0), (0,2), (2,0), (2,2), (2,3), (3,2), (3,3) }

This one is not reflexive

2. Relevant equations
I understand that reflective means a=a, but I don't understand how this one isn't. I think the real issue here is that I obviously don't understand exactly what reflexive really means.

Any help would be GREAT as I have an exam tomorrow morning and this is proving to be more difficult than I expected.

2. Jun 17, 2012

### vela

Staff Emeritus
For the relation R to be reflexive, you must have aRa for all a in {0, 1, 2, 3}.

3. Jun 17, 2012

### hammonjj

I apologize, but can you spell it out for me? I guess I don't understand why (1,1) is the problem, but not (1,0) and (0,1).

Thanks!

4. Jun 18, 2012

### vela

Staff Emeritus
Do you understand what the ordered pair (1,0) means in the context of relations?

5. Jun 18, 2012

### hammonjj

I think it means, in order to me an Equivalence Relation, there must also exist (0,1). Correct?

6. Jun 18, 2012

### vela

Staff Emeritus
Why would it mean that?

7. Jun 18, 2012

### algebrat

As an exercise, try finding the smallest set containing the above, which is also an equivalence relation. This idea, the completion of a set, is a pervasive one in advanced maths.

8. Jun 18, 2012

### vela

Staff Emeritus
Let $a, b \in X$ and $R \subset X\times X$. When you say $(a,b)\in R$, it means aRb, that is, a is related to b.

For a relation R to be reflexive, you must have that for every element a in X, aRa or, in ordered-pair notation, $(a,a) \in R$. Do you see now why your problem's R isn't reflexive?

9. Jun 18, 2012

### tonit

Reflexive doesn't mean a = a. The equality is a relation of equivalence, but a relation of equivalence need not be "=".

10. Jun 18, 2012

### HallsofIvy

Staff Emeritus
Reflexive means "if a is in the set, then (a, a) must be in the relation". 1 is in the set. Is (1, 1) in the relation?