Determining whether a relation is reflexive, symmetric, transitive.

Question

Let $X=\{0,1,2,...,10\}.$ Define the relation $R$ on $X$ by: for all $a,b$ in $X$, $a\mathrel{R}b$ if and only if $a+b=10$

is $R$ reflexive? symmetric? transitive?

$a\mathrel{R}a$
$a+a=10$
$2a=10$
$a=5$

Im confused, is that consider as reflexive?

Answer 1

You should easily be able to see that $R$ is symmetric, since if $a\:R\:b$ (that is, if $a+b=10$), then $b\:R\:a$ (that is, $b+a=10$).

In order to be reflexive on $X$, we must be able to say that $a\:R\:a$ (that is, $a+a=10$) for every $a\in X,$ which is not the case.

Note that the domain of $R$ is all of $X$. Given a symmetric transitive relation $S$ on a set $X,$ with the domain of $S$ being all of $X,$ we necessarily have that $S$ is reflexive on $X.$ (Why?) Consequently, $R$ cannot be transitive. More simply, one can find $a,b,c\in X$ such that $a+b=10$ and $b+c=10,$ but $a+c\ne 10,$ so $R$ is not transitive. (The elements $a,b,c$ will not all be the same, but they won't all be distinct, either.)

Answer 2

The property of reflexive is that $\forall x\in X, (x,x) \in R$. Clearly, $(1,1)\not\in R$, so it is not reflexive.