Give an example of a relation R on $A^2$ which is reflexive, symmetric, and not transitive

Question

I am just looking for some clarification on this exercise:

Let $A = \{a,b,c,d\}$. Give an example of a relation $R$ on $A^2$ which is reflexive, symmetric, and not transitive.

I understand that if I was only dealing with $A$ and and not $A^2$ I could give an answer similar to:

$R = \{(a,a),(a,b),(a,d),(b,a),(b,b),(c,c),(d,a),(d,d)\}$

Here $R$ is reflexive, symmetric, and not transitive.

What is confusing me is the fact that we have $A^2$. I am not sure what I am missing here. I am looking through my book to see if I can figure this out, but if someone could help shed some light on this it would be appreciated.

Thanks,

Tony

Answer 1

Based on this feed back I believe the answer is:

Let $A = \{a,b,c,d\}$, $A^2 = A \times A$ = $\{(a,a),(a,b),(a,c),(a,d),(b,a),(b,b),(b,c),(b,d),(c,a),(c,b),(c,c),(c,d),(d,a),(d,b),(d,c),(d,d)\}$

A relation $R$ on $A^2$ which is reflexive, symmetric, and not transitive is:

$R = \{(a,a),(a,b),(a,d),(b,a),(b,b),(c,c),(d,a),(d,d)\}$

R is reflexive because each element is related to itself.

R is symmetric because when one element is related to a second element that second element is related to the first.

R is not transitive because we have $bRa$ and $aRd$ but we do not have $bRd$.