Equivalence Relations

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.

vela

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

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!

vela

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

hammonjj

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

vela

Why would it mean that?

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.

vela

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?

tonit

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

HallsofIvy

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?