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

To determine which properties the relation \( R = \{ (A, B): A \cap B \neq \emptyset \} \) holds, we need to test it for reflexivity, symmetry, and transitivity. 

1. Reflexivity: A relation \( R \) on a set \( M \) is reflexive if every element in \( M \) is related to itself. Here, each subset \( A \) of \( S \), including the empty set, should satisfy \( A \cap A \neq \emptyset \). However, for any set \( A \), \( A \cap A = A \), which is always non-empty unless \( A \) is the empty set. Since the empty set is a subset of \( S \) and \( \emptyset \cap \emptyset = \emptyset \), the relation \( R \) is not reflexive as it fails for the empty set.
2. Symmetry: A relation \( R \) is symmetric if whenever \( (A, B) \in R \), then \( (B, A) \in R \). Here, if \( A \cap B \neq \emptyset \), then \( B \cap A = A \cap B \neq \emptyset \), satisfying the condition. Thus, the relation \( R \) is symmetric.
3. Transitivity: A relation \( R \) is transitive if whenever \( (A, B) \in R \) and \( (B, C) \in R \), then \( (A, C) \in R \). In our case, it is possible that \( (A, B) \in R \) and \( (B, C) \in R \), with \( A \cap C = \emptyset \). Thus, \( R \) is not transitive.

Given these analyses, the relation is symmetric only.

Thus, the correct answer is:

symmetric only

 

Answer 2

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.