I just want to brush up on my understanding of Relations with Sets. Specifically with this set:

$\{ 1, 2, 3 \}$

I understand Reflexive, Symmetric, Anti-Symmetric and Transitive in theory. But if it's not too much trouble, I'd like some help producing the appropriate R (relation) sets with the set above. So in a nutshell:

Question: What's the Relation sets for Reflexive, Symmetric, Anti-Symmetric and Transitive on the following set?:

$\{ 1, 2, 3 \}$


I've got some of them solved, but will complete my answer when I've figured out the rest.

Reflexive: $\{(1,1),(2,2),(3,3)\}$

Symmetric: $\{?\}$

Anti-Symmetric: $\{?\}$

Transitive: $\{(2,2),(2,3),(3,2),(3,3)\}$

  • $\begingroup$ There is not a unique relation with any of those properties on the given set. So what do you mean by the relation set? $\endgroup$ – Tobias Kildetoft Apr 22 '13 at 18:09
  • $\begingroup$ I mean just applying the properties of Reflexive, Symmetric, Anti-Symmetric and Transitive on the set shown above. $\endgroup$ – theCodeMonsters Apr 22 '13 at 18:10
  • 3
    $\begingroup$ But properties are not something you apply. They are something you test. And the above set is not a relation. $\endgroup$ – Tobias Kildetoft Apr 22 '13 at 18:11
  • $\begingroup$ No, that's what I'm trying to produce, see my edit. $\endgroup$ – theCodeMonsters Apr 22 '13 at 18:12
  • $\begingroup$ Are you looking for examples of relations with each of the mentioned properties, on the given set? $\endgroup$ – Tobias Kildetoft Apr 22 '13 at 18:13

First of all find $A\times A$ wherein $A=\{1,2,3\}$. So, we have $$A\times A=\{(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)\}$$ Now try to choose some subset of above set as relations on $A$, since we know that every relation on $A$ has the form $R\subseteq A\times A$. For example $R_1=\{(1,1),(2,2),(3,3)\}$ is reflexive. $R_2=\{(1,1),(1,2),(2,1),(2,2)\}$ is reflexive, transitive and symmetric. Now try to find other relations according to what you have learnt about them.

  • $\begingroup$ Good work, Babak, as usual $\ddot\smile\;\;+1\;\;$ $\endgroup$ – amWhy Apr 23 '13 at 0:32
  • $\begingroup$ @amWhy: Thanks Amy. :-) $\endgroup$ – Mikasa Apr 23 '13 at 1:33
  • $\begingroup$ $\Large \color{blue}{\bf\ddot\smile}\;\;$ $\endgroup$ – amWhy Apr 23 '13 at 1:40
  • $\begingroup$ You look great with that blue eyes. Anyway, I am going to the university. See you in the next hours Amy. Have a good peaceful sleep. $\endgroup$ – Mikasa Apr 23 '13 at 1:42
  • $\begingroup$ Have a wonderful day, Babak! $\endgroup$ – amWhy Apr 23 '13 at 1:49

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