Question

The minimum number of elements that must be added to the relation $R=\{(a, b),(b, c)\}$ on the set $\{ a , b , c \}$ so that it becomes symmetric and transitive is :

• A. 3
• B. 7
• C. 4
• D. 5

Answer 1

The Correct Option is B

To determine the minimum number of elements that must be added to the relation \( R = \{(a, b), (b, c)\} \) on the set \( \{ a, b, c \} \) so that it becomes symmetric and transitive, we need to follow these steps:

1. Definition of Symmetric: A relation \( R \) on a set is symmetric if, for every \( (x, y) \in R \), \( (y, x) \) is also in \( R \).
2. Definition of Transitive: A relation \( R \) on a set is transitive if, whenever \( (x, y) \in R \) and \( (y, z) \in R \), then \( (x, z) \) is also in \( R \).
3. Making the Relation Symmetric: Based on the symmetric property, we have:
   • For \( (a, b) \), add \( (b, a) \).
   • For \( (b, c) \), add \( (c, b) \).
   So, to make \( R \) symmetric, we add: \{ (b, a), (c, b) \} .
4. Making the Relation Transitive: Using the pairs in the symmetric relationship:
   • If \( (a, b) \) and \( (b, c) \) are in \( R \), we need \( (a, c) \).
   • If \( (b, a) \) and \( (a, b) \) are in \( R \), we need \( (b, b) \).
   • If \( (b, c) \) and \( (c, b) \) are in \( R \), we need \( (b, b) \).
   • If \( (c, b) \) and \( (b, c) \) are in \( R \), we need \( (c, c) \).
   • If \( (a, b) \) and \( (b, a) \) are in \( R \), we need \( (a, a) \).
   Therefore, to make \( R \) transitive, we additionally need: \{ (a, c), (b, b), (c, c), (a, a) \} .
5. Total Elements to Add:
   • Symmetric additions: \( \{ (b, a), (c, b) \} \)
   • Transitive additions: \( \{ (a, c), (b, b), (c, c), (a, a) \} \)
   • Total new pairs to add = \( 2 + 5 = 7 \)

Therefore, the minimum number of elements that must be added to \( R \) to make it both symmetric and transitive is 7.