Total number of equivalence relations defined in the set S = a, b, c is :

Question

Total number of equivalence relations defined in the set S = a, b, c is : (A) 5 (B) 3! (C) 23 (D) 33

Tardigrade Admin · Accepted Answer

The smallest equivalence relation is the identity relation

R_{1} =

< b r / > {(a, a), (b, b), (c, c)} < b r / >

Then two ordered pairs of two distinct elements can be added to give three more equivalence relations.

< b r / > R_{2} = {(a, a), (b, b), (c, c), (a, b), (b, a)} < b r / >

Similarly

R_{3}

and

R_{4}

can be made by taking

(b, c), (c, b)

and (a, c), (c, a) respectively.

Finally the largest equivalence relation i.e., the universal relation

< b r / > R_{5} = {(a, a), (b, b), (c, c), (a, b), (b, a), (a, c), (c, a), (b, c), (c, b)} < b r / >

Hence, total 5 equivalence relations can be created.

Q. Total number of equivalence relations defined in the set S = {a, b, c } is :