Question

Let \( A = \{a, b, c\} \), then the number of equivalence relations on \( A \) containing \( (b, c) \) is:

• A. 3
• B. 2
• C. 4
• D. 1

Hint:

In equivalence relations, the relation must be reflexive, symmetric, and transitive. Always check if the given pairs satisfy these properties.

Answer 1

The Correct Option is B

For the set \( A = \{a, b, c\} \), we need to find the number of equivalence relations containing \( (b, c) \). An equivalence relation must be reflexive, symmetric, and transitive. Let’s find all the possible equivalence relations: 1. \( R_1 = \{(a, a), (3, b), (c, c), (b, c), (c, b)\} \) 2. \( R_2 = \text{universal relation} = \{(a, a), (b, b), (c, c), (b, c), (c, b)\} \) Thus, there are 2 equivalence relations that contain \( (b, c) \). Thus, the number of equivalence relations is 2.