Help understanding what's required for a relation to be symmetric?

Question

Let $\,X=\{1,2,3,4,5\},\;$ does a symmetric relation on $\,X\,$ need to have all the elements of $\,X\,$ in the relation?

Or can if have just a few elements of $X$ like this relation: $\,A =\{(1,2),(2,1)\}\;?\,$

I would appreciate your help

Answer 1

$\rho$ be a relation on $X$ i.e $\rho\subseteq X\times X$, $\rho$ is symmetric if $(x,y)\in\rho\Rightarrow (y,x)\in \rho$ i.e $x\rho y\Rightarrow y\rho x$

Answer 2

For symmetry to hold for a relation $R$ on a set $X$, we need

$$\forall a,b\in X,\ a R b\iff b R a$$

The "vacuous truth" of the empty relation, or (for example) of one where only one double-pair $(a,b),(b,a)$ is in the relation suffices as a symmetric relation. As long as the condition is satisfied that "whenever $a$ relates to $b$, then $b$ relates to $a$," then a relation is symmetric.

Answer 3

The relation you post is symmetric. What we need for symmetry is the following: For every pair $(x, y) \in A$, $(y, x)$ is also in $R$. However, if $(x, y) \notin R$ then $(y, x)$ need not be in $A$.

So IF $(x, y) \in A$, then we must have $(y, x) \in A$. That doesn't tell us that $(x, y), (y, x)$ must be in $R$, only that if $(x, y)\in A$, then we must have $(y, x) \in A$.

And your relation $A$ on $X$ meets that condition. We have that $(1, 2) \in A$. So therefore for symmetry, we need that $(2, 1) \in A$, and vice versa. To be a symmetric relation doesn't require that every possible pair of elements in $X$ must be related symmetrically.