Diagonal relation and reflexive relation

Question

Let R be a non trivial relation on set X . If R is symmetric and anti symmetric then R is a ) reflexive b ) transitive c ) equivalence d ) diagonal relation Actually I am confused about definition of diagonal relation i.e whether diagonal relation contains ( x ,x ) for every x belongs to Set or it contains ( x ,x ) for some x belongs to Set . And whether there is any difference between reflexive and diagonal relation ?

Answer 1

Given that $R$ is simultaneously symmetric and antisymmetric, it follows that $R\subseteq\{(x,x)~:~x\in X\}\subseteq X\times X$

To see why this is:...

suppose for contradiction that there was in fact some $(x,y)\in R$ where $x\neq y$. By symmetry this implies that $(y,x)\in R$ as well. By antisymmetry this implies that $(y,x)\notin R$. These two observations contradict eachother. This implies that there could not possibly be any pair $(x,y)\in R$ where $x\neq y$.

A reflexive relation is one where $\{(x,x)~:~x\in X\}\subseteq R$. That is to say, it contains every possible pair of the form $(x,x)$ along with possibly containing other types of pairs too.

A diagonal relation is one where $R\subseteq\{(x,x)~:~x\in X\}$. That is to say, it contains only pairs of the form $(x,x)$, possibly not all of them, and contains no other pairs of a different form.

It is possible for a diagonal relation to be reflexive and vice versa, but only in the case that the relation is specifically $R=\{(x,x)~:~x\in X\}$

We have as a result:

Regarding reflexivity:

$R$ is not necessarily reflexive, but could be

Regarding transitivity:

$R$ is transitive. Since the only way that $(a,b)$ and $(b,c)$ are elements of the relation is if $a=b=c$, it clearly follows that $(a,c)$ is also in the relation.

Regarding equivalence:

As $R$ is not necessarily reflexive, it is again not necessarily an equivalence relation.

Regarding diagonality:

$R$ is a diagonal relation