Question

Let \( X = {R} \times {R} \). Define a relation \( R \) on \( X \) as: \[ (a_1, b_1) \, R \, (a_2, b_2) \iff b_1 = b_2. \] Statement-I: \( R \) is an equivalence relation. Statement-II: For some \( (a, b) \in X \), the set \( S = \{(x, y) \in X : (x, y) R (a, b)\ \) represents a line parallel to \( y = x \).}

• A. Both Statement-I and Statement-II are false.
• B. Statement-I is true but Statement-II is false.
• C. Both Statement-I and Statement-II are true.
• D. Statement-I is false but Statement-II is true.

Hint:

To check if a relation is an equivalence relation, verify reflexivity, symmetry, and transitivity. Also, when interpreting geometric shapes, note the difference between lines parallel to \( y = x \) and horizontal lines.

Answer 1

The Correct Option is B

We are given the relation \( R \) on \( X = {R} \times {R} \) defined by: \[ (a_1, b_1) \, R \, (a_2, b_2) \iff b_1 = b_2. \] Statement-I: \( R \) is an equivalence relation.
To verify if \( R \) is an equivalence relation, we check the following properties: - Reflexive: Since \( b_1 = b_1 \) for all \( (a, b) \), \( R \) is reflexive. - Symmetric: Since \( b_1 = b_2 \) implies \( b_2 = b_1 \), \( R \) is symmetric. - Transitive: Since \( b_1 = b_2 \) and \( b_2 = b_3 \), it follows that \( b_1 = b_3 \), so \( R \) is transitive. Thus, \( R \) is an equivalence relation, so Statement-I is true. Statement-II: For some \( (a, b) \in X \), the set \( S = \{(x, y) \in X : (x, y) R (a, b)\ \) represents a line parallel to \( y = x \).}
We are given that \( (x, y) R (a, b) \) implies \( b = y \). This describes a horizontal line at \( y = b \), not a line parallel to \( y = x \). Thus, Statement-II is false.