While solving questions related to reflexivity, symmetricity and transivity of relations, I came across this question:
Show that the relation $R$ in the set $A = \{1,2,3\}$ given by $R = \{(1,2),(2,1)\}$ is symmetric, but neither reflexive, not transitive.
How is this relation symmetric? A symmetric relation is defined on Wikipedia as follows : a binary relation $R$ over a set $X$ is symmetric if it holds for all $a$ and $b$ in $X$ that if $a$ is related to $b$ then $b$ is related to $a$.
In the relation in question, shouldn't the elements $(1,3),(3,1),(2,3),(3,2)$ also be present to account for all $a$,$b\in$A.