Equivalence Relations on {0, 1, 2, 3}: Understanding Reflexivity and Properties

[hammonjj]

Homework Statement

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

Homework 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]

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

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]

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

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]

Homework Statement

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

Homework 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.

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]

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.

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?