Question

Let \( S = \{ 1, 2, 3, \ldots, 10 \} \). Suppose \( M \) is the set of all the subsets of \( S \), then the relation \( R = \{ (A, B): A \cap B \neq \phi; A, B \in M \} \) is:

• A. symmetric and reflexive only
• B. reflexive only
• C. symmetric and transitive only
• D. symmetric only

Answer 1

The Correct Option is D

Let’s analyze the properties of the relation \(R\).

Step 1. Reflexivity: For reflexivity to hold, each subset \( A \) in \( M \) should satisfy \( A \cap A \neq \emptyset \). Since \( A \cap A = A \), \( R \) would be reflexive if \( A \neq \emptyset \) for every \( A \in M \). However, the empty set \( \emptyset \in M \) does not satisfy \( \emptyset \cap \emptyset \neq \emptyset \), so \( R \) is not reflexive.

Step 2. Symmetry: If \( (A, B) \in R \), then \( A \cap B \neq \emptyset \). This implies \( B \cap A \neq \emptyset \), so \( (B, A) \in R \). Therefore, \( R \) is symmetric.

Step 3. Transitivity: Suppose \( (A, B) \in R \) and \( (B, C) \in R \), meaning \( A \cap B \neq \emptyset \) and \( B \cap C \neq \emptyset \). However, \( A \cap C \) may still be empty, so \( R \) is not transitive.

Thus, the relation \( R \) is symmetric only.