JEE MAIN - Mathematics (2023 - 24th January Evening Shift - No. 15)
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 __________.
Answer
13
Explanation
$R=\{(a, b)(b, c)(b, d)\}$
$S:\{a, b, c, d\}$
Adding $(a, a),(b, b),(c, c),(d, d)$ make reflexive.
Adding $(b, a),(c, b),(d, b)$ make Symmetric
And adding $(a, d),(a, c)$ to make transitive
Further $(d, a) \&(c, a)$ to be added to make Symmetricity.
Further $(c, d) \&(d, c)$ also be added.
So total 13 elements to be added to make equivalence.
$S:\{a, b, c, d\}$
Adding $(a, a),(b, b),(c, c),(d, d)$ make reflexive.
Adding $(b, a),(c, b),(d, b)$ make Symmetric
And adding $(a, d),(a, c)$ to make transitive
Further $(d, a) \&(c, a)$ to be added to make Symmetricity.
Further $(c, d) \&(d, c)$ also be added.
So total 13 elements to be added to make equivalence.
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