Let R be the relation {(0, 1), (1, 1), (1, 2), (2, 0), (3, 0)} defined on the set {0, 1, 2, 3}. Find the following:
a.reflexive closure of R. b.symmetric closure of R. c.The transitive closure of R.
The reflexive closure of R is the relation containing the ordered pairs (0,0), (0,1), (1, 1), (1,2), (2,0), (3,0), (3,3) The symmetric closure of R is the relation containing the ordered pairs (0,1), (0,2), (0,3), (1,0), (1,1), (1,2), (2,1), (3,0) The transitive closure of R is the relation containing the ordered pairs (0, 1), (0,2), (1,1), (1,2), (2,0), (2,1), (3,0), (3,1), (3,2)
Is this correct?
Consider a relation $R$. We want to add all edges to make $R$ reflexive. Hence, $R' = R \cup ${$(x,y)$}. This relation $R'$ is the reflexive closure of $R$. We add all pairs $(a,a)$ that does not exist. Hence, the answer is obtained by adding these elements: $(0,0),(2,2),(3,3)$ to $R$.
In order to find the symmetric closure of a relation $R$, we add an edge from $a$ to $b$, where there is already an edge from $b$ to $a$. The symmetric closure of $R$ is $R \cup R^{-1}$,i.e, if $R $={$(a,b)\mid ... $}, then $R^{-1}$ ={$(b,a)|...$}. We add all pairs of edges $(a,b)$ where $(b,a)$ exists.
In order to find the transitive closure of a relation $R$, we add an edge from $a$ to $c$, when there are edges from $a$ to $b$ and $b$ to $c$. Hence, you can check your answer on these lines. Hope it helps.