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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), and (3, 0). Find the a) reflexive closure of R. b) symmetric closure of R.

Solution

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DEFINITIONS

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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