# Is a relation that is purely reflexive also symmetric?

Is a relation that is pruely reflexive also symmetric?

For example, say you have a relation defined as $$R = \{(a,a),(b,b)\}$$. This is purely reflexive, but is it also symmetric? The typical symmetric definition is $$aRb \Leftrightarrow bRa$$, which is kinda shown in this as $$aRa \Leftrightarrow aRa$$, but I am unsure. Sorry if this is a trivial question, I am just learning about this stuff in a proof course and am slightly confused.

• Yes, it’s symmetric, because it is certainly true that $a\,R\,a\leftrightarrow a\,R\,a$. Jul 26, 2020 at 2:10
• Yes. You need to show that, if $x \, R \, y$, then $y \, R \, x$ to show symmetry. In such relations, if $x \, R \, y$, then we must have $x = y$, and so $x \, R \, y$ means the same as $x \, R \, x, y \, R \, x,$ and $y \, R \, y$. Jul 26, 2020 at 2:29

For example, if a relation is reflexive, the diagonal elements will all be populated with 1's. Note that a "1" indicates that the corresponding row entry and column entry is in the relation, and a "0" means that combination in not in the relation. For example, $$\left(a,a\right) \in R$$, but $$\left(a,b\right) \notin R$$, etc. in the above table.