# Relational Sets for Reflexive, Symmetric, Anti-Symmetric and Transitive

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

• There is not a unique relation with any of those properties on the given set. So what do you mean by the relation set? – Tobias Kildetoft Apr 22 '13 at 18:09
• I mean just applying the properties of Reflexive, Symmetric, Anti-Symmetric and Transitive on the set shown above. – theCodeMonsters Apr 22 '13 at 18:10
• But properties are not something you apply. They are something you test. And the above set is not a relation. – Tobias Kildetoft Apr 22 '13 at 18:11
• No, that's what I'm trying to produce, see my edit. – theCodeMonsters Apr 22 '13 at 18:12
• Are you looking for examples of relations with each of the mentioned properties, on the given set? – 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.
• Good work, Babak, as usual $\ddot\smile\;\;+1\;\;$ – amWhy Apr 23 '13 at 0:32
• $\Large \color{blue}{\bf\ddot\smile}\;\;$ – amWhy Apr 23 '13 at 1:40