Question

Let  R = {(1, 2), (2, 3), (3, 3)} be a relation defined on the set \( \{1, 2, 3, 4\} \). Then the minimum number of elements needed to be added in \( R \) so that \( R \) becomes an equivalence relation, is:}

• A. \( 10 \)
• B. \( 8 \)
• C. \( 9 \)
• D. \( 7 \)

Hint:

For a relation to be an equivalence relation, make sure it satisfies reflexivity, symmetry, and transitivity. Adding pairs systematically helps to meet these conditions.

Answer 1

The Correct Option is D

Step 1: Identifying the properties of equivalence relations. For a relation to be an equivalence relation, it must be reflexive, symmetric, and transitive. Step 2: Making the relation reflexive. For reflexivity, every element of the set must relate to itself. So we need to add the pairs: \[ (1, 1), (2, 2), (3, 3), (4, 4). \] Now, the relation contains all reflexive pairs. Step 3: Making the relation symmetric. To make the relation symmetric, for every pair \( (a, b) \), we need to add the pair \( (b, a) \) if it is not already present. Therefore, we add: \[ (2, 1), (3, 2), (3, 1), (1, 3). \] Step 4: Total elements to be added. The total number of elements added is: \[ (1, 1), (2, 2), (3, 3), (4, 4), (2, 1), (3, 2), (3, 1), (1, 3). \] Thus, the minimum number of elements to be added is 7.