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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.

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

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  • $\begingroup$ 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. $\endgroup$ – agdhruv Jan 26 '16 at 17:47
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    $\begingroup$ @ag_dhruv yes, reflexivity requires for all $a\in A$ $\endgroup$ – user160738 Jan 26 '16 at 18:42

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