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I'm studying relations at the moment, but I can't solve this problem. Could you please give me a piece of advice? I really want to understand how to solve it. Thank you in advance!

Let R be a relation on a nonepty set X. Is this relation reflexive, transitive, symmetric, antisymmetric? Is it equivalence relation, partial order, total order? If it's an equivalence relation, identify the equivalence classes.

  1. X = {1,2,3,4,5,6,7,8,9}, R = {(a,b)| a + b is even number},

  2. X = $\mathbb R^2$, R = "symmetric with respect to x-axis",

  3. X =$\mathbb R^2$, R = "is at the same distance form origin",

  4. X = {all human beings}, R = "to be brother",

  5. X = {all human beings}, R = "live in the same city".

Here are my thoughts on this so far (I may be really wrong, but anyway):

  1. Reflexive and symmetric.
  2. Symmetric.
  3. Reflexive and symmetric.
  4. Antisymmetric?
  5. Also antisymmetric?
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2 Answers 2

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5) is symmetric (if x lives in same city as y, then y lives in same city as x).

5) is also reflexive (x lives in same city as x), but 4 is not (no one is brother of themselves .. so 4 is in fact irreflexive).

4) is not antisymmetric: x and y can be brothers of each other. Also not symmetric (x can be brother of y, but y can be sister of x)

Any thoughts on transitivity for these?

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  • $\begingroup$ Thank you so much! I guess I see it now... As for transitivity, I'm almost sure that 4th can't be transitive. $\endgroup$
    – user497449
    Commented Oct 30, 2017 at 17:26
  • $\begingroup$ @user497449 .. and why not? What's your explanation? $\endgroup$
    – Bram28
    Commented Oct 30, 2017 at 17:27
  • $\begingroup$ I was thinking... If x is related to y and y is related to z, then x is related to z, right? But if the relation is "to be brother", it doesn't have to be true, because x and z may not be siblings. I'm not sure if it's correct though. $\endgroup$
    – user497449
    Commented Oct 30, 2017 at 17:30
  • $\begingroup$ @user497449 well, if x is brother of y, and y brother of z ... then x is definitely male, and so it seems as if x would be brother of z (also because we don;t care about gender of z) .. But here is the thing: x and z could be the same person! $\endgroup$
    – Bram28
    Commented Oct 30, 2017 at 17:32
  • $\begingroup$ Oh, I get it! Thank you for explanation!! $\endgroup$
    – user497449
    Commented Oct 30, 2017 at 17:33
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a and b live in the same city when there is some city in which both a and b live.

Joe is a farmer who lives on his farm outside of any city.
Joe is in X. There is no city in which both Joe and Joe live.

Thus 5 is not an equivalence relation because it is not reflexive.

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