Conditions for a relation to be symmetric.

Question

Let $A = \{1, 2, 3\}$, and $R$ be a relation on $A$. $R = \{(1, 1), (2, 2), (3, 3)\}$

Is $R$ a symmetric relation? Since for all $a R b$ there is $b R a$. If $R$ is not symmetric, then does that mean for a relation to be symmetric $a$ must not equal to $b$ ?

Answer 1

$R$ is symmetric because $aRb \implies bRa$ for all $a,b \in A$.

As for your other query, no, nothing like that is needed. In fact, such a question desn't even arise if $a=b$.

Answer 2

$R$ is a symmetric relation means if $(a,b)\in R$, then $(b,a)\in R$. Elements of $R$ are $(1, {\color{red} 1}),(2, {\color{red} 2})$ and $(3,{\color{red} 3})$.

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Now, you must check that $({\color{red} 1},1 )$,$({\color{red} 2},2 )$ and $({\color{red} 3},3 )$ are elements of $R$.