Let R = {(3,3),(6,6),(9,9),(12,12),(6,12),(3,9), (3,12),(3,6)} be a relation on the set A = { 3,6,9,12}. The relation is: Relation (Mathematics)

(d) : For $(3,9) \in R,(9,3) \notin R$ $\\ \therefore$ Relation is not symmetric which means our choice (a) and (b) are out of court. $\\$ We need to prove reflexivity and transitivity. $\\$ For reflexivity $a \in R,(a,a) \in R$ which is hold i.e. $R$ is reflexive. $\\$ Again, for transitivity of $(a,b) \in R,(b,c) \in R$ $\\ \Rightarrow (a,c) \in R$ which is also true in $\\R = \{(3,3),(6,6),(9,9),(12,12),(6,12)$ , $(3,9),(3,12),(3,6)\}$ .

Explanation

The minimum number of elements that must be added to the relation R = {(a,b),(b,c),(b,d)} on the set { a,b,c,d} so that it is an equivalence relation is

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