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Question

# Let R be the relation on the set {0, 1, 2, 3} containing the ordered pairs (0, 1),(1, 1),(1, 2),(2, 0),(2, 2),(3, 0). Find reflexive, symmetric and transitive closure of R.

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DEFINITIONS

A relation $R$ on a set $A$ is $\textbf{reflexive}$ if $(a,a)\in R$ for every element $a\in A$.

A relation $R$ on a set $A$ is $\textbf{symmetric}$ if $(b,a)\in R$ whenever $(a,b) \in R$

A relation $R$ on a set $A$ is $\textbf{transitive}$ if $(a,b)\in R$ and $(b,c) \in R$ implies $(a,c)\in R$

The $\textbf{reflexive closure}$ of $R$ is the relation that contains all ordered pairs of $R$ and to which all ordered pairs of the form $(a,a)\in R$ ($a\in A$) were added (when they were not present yet).

$R\cup \Delta=R\cup \{(a,a)|a\in A\}$

The $\textbf{symmetric closure}$ of $R$ is the union of the relation $R$ with its inverse relation $R^{-1}$.

The $\textbf{inverse relation}$ $R^{-1}$ is the set $\{(b,a)|(a,b)\in R\}$

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