# Relation is reflexive but not symmetric, how to motivate?

My teacher in discrete math is very meticulous about how we motivate our answers in the course.

For instance, the exercise:

Put A={1,2,3} and form the relation R on A by putting R= {(1, 1),(2, 2),(3, 3),(1, 2),(2, 3),(3, 1)}.

Investigate if R is: reflexive and symmetric. If the relation has a property, give proof for it and if the relation doesn't have the property, prove it.

It is reflexive: A is {1,2,3} and in R there is (1,1), (2,2), and (3,3). Every element in A exists as a pair in R, therefore it is reflexive.

It is not symmetric, as in R there is (1,2), (2,3) and (3,1), but there is no (2,1), (3,2) or (1,3) respectively. For symmetry, every pair in R would have needed a reserve pair.

Am I correct in my motivations or is there a better way to motivate myself?

• Did you mistype $R$'s contents? Because you claimed $(3,\,3)\notin R$. – J.G. Feb 16 '20 at 11:46
• Yes sorry, (3,3) is in R so it is reflexive – Jean Doe Feb 16 '20 at 11:56
• Strictly speaking, you just need one counterexample for symmetry. $(1,2)$ would do as $(1,2)\in R$ and $(2,1)\not\in R$. No harm done if you mention more counterexamples, but it is not necessary. – Stinking Bishop Feb 16 '20 at 12:03
• Yes that is true. I'm more anxious about having motivated myself good enough at the reflexivity part. How would you motivate that this relation is reflexive? – Jean Doe Feb 16 '20 at 12:04