Let R=(1,2),(2,3),(3,3) be a relation defined on the set 1,2,3,4.

Question

Let $R={(1,2),(2,3),(3,3)}$ be a relation defined on the set ${1,2,3,4}$. Then the minimum number of elements, needed to be added in R so that R becomes an equivalence relation, is:

Accepted Answer

For reflexivity, add $(1,1),(2,2),(4,4)$ --- 3 pairs (since $(3,3)$ already present).
For symmetry, add $(2,1),(3,2)$ --- 2 pairs.
For transitivity: $(1,2),(2,3)\Rightarrow(1,3)$ required, and by symmetric closure also $(3,1)$ --- 2 more pairs.

Total pairs added: $(1,1),(2,2),(4,4),(2,1),(3,2),(1,3),(3,1)$ --- 7 elements.

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