Question

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

• A. 5
• B. 3!
• C. $2^3$
• D. $3^3$

Answer 1

Answer: (A)

5

Answer 2

The smallest equivalence relation is the identity relation $R_1$ = {(a, a), (b, b), (c, c) Then two ordered pairs of two distinct elements can be added to give three more equivalence relations $R_2$ = {(a, a), (b, b), (c, c), (a, b), (b, a)} Similarly $R_3$ and $R_4$. Finally the largest equivalence relation i.e, the universal relation $R_5$ = { (a, a), (b, b), (c, c), (a, b), (b, a), (a, c), (c, a), (b, c), (c, b)}