understanding reflexive transitive closure

Question

Suppose I have the following relation $$R = \{(1,1), (2,3), (3,1)\}$$ To make it reflexive we add the following missing pairs: $$ \{(2,2), (3,3)\}$$ Now I wonder how to find the reflexive transitive closure of that relation?

Answer 1

Reflexive and transitive are two different things.
Transitive means: if $a \sim b$ and $b \sim c$ then $a \sim c$. While reflexive means that $a\sim a$ in your realtion.

You have $R = \{(1,1), (2,3), (3,1)\}$ and you want it to be closed under transitivity . Since $2\sim 3$ and $3\sim 1$ you must have $2\sim 1$ so you add $(2,1)$.

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Edit:

Suppose that $R$ is a relation on a set $A$. The reflexive transitive closure of $R$ is the smallest relation $S$ on $A$ such that:

$R⊆S$;

$S$ is reflexive;

and $S$ is transitive.