Question

Let \( S \) denote the set of all subsets of integers containing more than two numbers. A relation \( R \) on \( S \) is defined by:

\[ R = \{ (A, B) : \text{the sets } A \text{ and } B \text{ have at least two numbers in common} \}. \]

Then the relation \( R \) is:

• A. reflexive, symmetric and transitive
• B. reflexive and symmetric but not transitive
• C. not reflexive, not symmetric and not transitive
• D. not reflexive but symmetric and transitive
• E. reflexive but not symmetric and transitive

Hint:

When checking properties of relations, carefully examine whether the relation satisfies the conditions for reflexivity, symmetry, and transitivity.

Answer 1

The Correct Option is B

The given relation \( R \) is defined as: for two sets \( A \) and \( B \), \( (A, B) \in R \) if and only if \( A \) and \( B \) share at least two elements.
Let's check the properties of the relation \( R \):
1. Reflexivity:
For a set \( A \), \( (A, A) \in R \) if \( A \) has at least two elements. Since \( A \) and itself will always share at least two elements if \( |A| \geq 2 \), the relation is reflexive.
2. Symmetry:
If \( (A, B) \in R \), then \( A \) and \( B \) have at least two elements in common. Since the relationship between \( A \) and \( B \) is symmetric, \( (B, A) \in R \) as well. Therefore, the relation is symmetric.
3. Transitivity:
For transitivity to hold, if \( (A, B) \in R \) and \( (B, C) \in R \), then we must have \( (A, C) \in R \). However, this is not always true. For example, if \( A = \{1, 2, 3\} \), \( B = \{2, 3, 4\} \), and \( C = \{3, 4, 5\} \), we have \( (A, B) \in R \) and \( (B, C) \in R \), but \( (A, C) \notin R \) because \( A \) and \( C \) only share one element, not two. Therefore, the relation is not transitive.
Since the relation is reflexive and symmetric but not transitive, the correct answer is (B).