# Is this relation symmetric?

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.

• No: Symmetry implies, e.g., that since $(1, 3)$ is not in the relation, neither is $(3, 1)$. – Travis Willse Jan 26 '16 at 17:37

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 $$R$$.

• Ah, I get it. Just a clarification: a relation is reflexive only if $(a,a)$ is present in the relation for all $a \in A$, right? So that, if $A = \{1,2,3\}$ and $R=\{(1,1),(2,2)\}$, $R$ is not reflexive until and unless $\{(3,3)\}$ is not present in the relation. – agdhruv Jan 26 '16 at 17:47
• @ag_dhruv yes, reflexivity requires for all $a\in A$ – user160738 Jan 26 '16 at 18:42