Is this relation symmetric?

Question

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.

Answer 1

Why would, for example $(1,3)$ need to be in the relation?

The definition is:

$R$ is symmetric if (and only if) it holds for all $a$ and $b$ that if $(a,b)\in R$ then $(b,a)\in R$

which we can unfold to

$R$ is symmetric if (and only if) all of the following are true:

• If $(1,1)\in R$ then $(1,1)\in R$
• If $(1,2)\in R$ then $(2,1)\in R$
• If $(1,3)\in R$ then $(3,1)\in R$
• If $(2,1)\in R$ then $(1,2)\in R$
• If $(2,2)\in R$ then $(2,2)\in R$
• If $(2,3)\in R$ then $(3,2)\in R$
• If $(3,1)\in R$ then $(1,3)\in R$
• If $(3,2)\in R$ then $(2,3)\in R$
• If $(3,3)\in R$ then $(3,3)\in R$

The only of these conditions that even mention $(1,3)$ are

• If $(1,3)\in R$ then $(3,1)\in R$
• If $(3,1)\in R$ then $(1,3)\in R$

and they are both satisfied because neither $(1,3)$ nor $(3,1)$ are in $R$.