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I'm doing exercises of equivalence relations. Reviewing the material I have in hand I get this definition:

A relation R in a Set S is called equivalence relation if R es reflexive, symmetric and transitive.

I understand the properties in numbers but in the material is applied to words and I don't know if I understand well. I would appreciate your help.

The examples are:

Indicate which of the following relationships are reflexive, symmetric or transitive on the set of all people:

a) "is older than" - transitive

So let me see if I'm getting it. Let's say the person is Mary. So it is not reflexive because Mary can't be older than herself. It is not symmetric because Mary can be older than Will but then Will can't be older than Mary. It is transitive because Mary can be older than Will and Will older than Peter, and then Mary is older than James.

b) "sits in the same row as" - reflexive, symmetric, transitive. This is an equivalence relation.

How is reflexive? Does Mary sit in the same row as herself? That does not make sense. It is symmetric because if Mary sits in the same row as Will, then Will sits in the same as Mary. It is transitive because if Mary sits in the same row as Will and Will sits in the same row as Peter, then Mary sits in the same row as Peter.

c) "weighs more than" - transitive

Is think is transitive for the same reason as the a).

I have to apply that to these exercises:

d) "is the sister/brother of" - If is that way I think it is applied then this is just symmetric?

Mary can't be the sister of herself. So it is not reflexive. If Mary is sister of Will, then Will is brother of Mary. So it is symmetric. Then I don't think it could be transitive because Mary can be sister of Will and Will brother of Peter, but Mary might not be necessary sister of Peter.

e) "is the same height of"

reflexive, symmetry and transitive?

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a) "is older than" - transitive

Everything is right here.

b) "sits in the same row as" - reflexive, symmetric, transitive. This is an equivalence relation.

Same here. To answer a question, does Mary sit in the same row as herself? Of course, because she can't sit in two different rows simultaneously.

c) "weighs more than" - transitive

Bingo.

d) "is the sister/brother of" - If is that way I think it is applied then this is just symmetric?

Yeah this is a weird one because of the inherent gendered language. Personally, I would interpret it was instead "is the sibling of." In which case you would get transitivity as well.

e) "is the same height of"

reflexive, symmetry and transitive?

Yup. Since you appear to be unsure:

  • Mary is the same height as herself, trivially, so reflexive.
  • If Mary is the same height as John, John is as tall as her, so symmetric.
  • If Mary is the same height as John, and John the same as Jill, then Mary is the same height as Jill.

If you were to quantify heights, this would not be unlike the equality relation, i.e. $a=a$, and $a=b$ iff $b=a$, and $a=b$ and $b=c$ imply $a=c$. In fact, all of the above might be more obvious if you envision a particular height for Mary and follow the chain of implications.

Overall I'd say you appear to understand this material just fine!

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