If R is the smallest equivalence relation on the set 1, 2, 3, 4...

R

is the smallest equivalence relation on the set

{ 1, 2, 3, 4 }

such that

{ ( 1, 2 ) , ( 1, 3 ) } ⊂ R

, then the number of elements in

R

is ______.

Options

A $10$
B $12$
C $8$
D $15$

Correct answer

A. $10$

Step-by-step solution

Given: $R = \left\{\left(1, 3\right), \left(1, 2\right)\right\}$

To make $R$ as Reflexive, the required elements are \left(1, 1\right), \left(2, 2\right), \left(3, 3\right)\text{and}\left(4, 4\right) $To make$ R $as Symmetric, the required elements are \left(3, 1\right)\text{and}\left(2, 1\right)$

To make $R$ as Transitive, the required elements are \left(3, 2\right)\text{and}\left(2, 3\right) $So,$ 8 $elements are required to make$ R $an equivalence relation.Hence, the total number of elements in$ R $is$ 10$.

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