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

To make the relation \( R = \{(1, 2), (2, 3), (3, 3)\} \) an equivalence relation on the set \( \{1, 2, 3, 4\} \), we need to ensure it satisfies three properties: reflexivity, symmetry, and transitivity.

1. Reflexivity: Each element must be related to itself. Therefore, we must add the pairs: \((1,1)\), \((2,2)\), \((4,4)\).

2. Symmetry: If \((a, b)\) is in the relation, then \((b, a)\) must also be in it. For existing pairs, add: \((2,1)\), \((3,2)\).

3. Transitivity: If \((a, b)\) and \((b, c)\) are in the relation, then \((a, c)\) must also be in it. Evaluate existing pairs:

• \((1,2)\) and \((2,3)\) imply \((1,3)\)
• Using the new pair \((3,2)\) and \((2,1)\), infer \((3,1)\)

Now, enumerating all added pairs, we find: \((1,1)\), \((2,2)\), \((4,4)\), \((2,1)\), \((3,2)\), \((1,3)\), \((3,1)\). Therefore, 7 elements are added in total.

Conclusion: The minimum number of elements to be added is \(7\).