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

To determine the truth of the statements regarding the relation \( R \) defined on \( X = \mathbb{R} \times \mathbb{R} \), let's analyze each statement:

1. Statement-I: \( R \) is an equivalence relation.
   • Reflexivity: For any pair \((a, b) \in X\), we need \((a, b) \, R \, (a, b)\). According to the definition of \( R \), this means \(b = b\), which is true. Therefore, \( R \) is reflexive. 
   • Symmetry: For any \((a_1, b_1), (a_2, b_2) \in X\), if \((a_1, b_1) \, R \, (a_2, b_2)\), then by definition \(b_1 = b_2\). For symmetry, we need \((a_2, b_2) \, R \, (a_1, b_1)\), which also requires \(b_2 = b_1\). Since equality is symmetric, \(R\) is symmetric.
   • Transitivity: For any \((a_1, b_1), (a_2, b_2), (a_3, b_3) \in X\), if \((a_1, b_1) \, R \, (a_2, b_2)\) and \((a_2, b_2) \, R \, (a_3, b_3)\), this implies \(b_1 = b_2\) and \(b_2 = b_3\). Therefore, \(b_1 = b_3\), ensuring that \((a_1, b_1) \, R \, (a_3, b_3)\), so \(R\) is transitive.
2. 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 \(y = x\).

Conclusion: From the analysis above, we conclude that Statement-I is true but Statement-II is false.

Answer 2

To determine the correctness of the given statements, we start by analyzing the definition and properties of the relation \( R \) on the set \( X = \mathbb{R} \times \mathbb{R} \), where two pairs \((a_1, b_1)\) and \((a_2, b_2)\) are related, i.e., \((a_1, b_1) \, R \, (a_2, b_2)\), if and only if \( b_1 = b_2 \).

Statement-I: Verification of Equivalence Relation 

An equivalence relation is defined as a relation that is reflexive, symmetric, and transitive. Let's check each property:

• Reflexive: For any \((a, b) \in X\), it holds that \( (a, b) \, R \, (a, b) \) because \( b = b \). Thus, the relation is reflexive.
• Symmetric: If \((a_1, b_1) \, R \, (a_2, b_2)\), then \( b_1 = b_2 \). Therefore, \((a_2, b_2) \, R \, (a_1, b_1)\) because \( b_2 = b_1 \). Hence, the relation is symmetric.
• Transitive: If \((a_1, b_1) \, R \, (a_2, b_2)\) and \((a_2, b_2) \, R \, (a_3, b_3)\), then \( b_1 = b_2 \) and \( b_2 = b_3 \), which implies \( b_1 = b_3 \). Thus, \((a_1, b_1) \, R \, (a_3, b_3)\), confirming that the relation is transitive.

Since \((a_1, b_1) \, R \, (a_2, b_2)\) is reflexive, symmetric, and transitive, Statement-I is true.

Statement-II: Analyzing the Set \( S \)

The set \( S \) is defined as: \(S = \{(x, y) \in X : (x, y) \, R \, (a, b)\} = \{(x, y) \in X : y = b\}.\)

This defines a horizontal line in the plane \( y = b \). A line given by \( y = b \) is parallel to the x-axis and is not parallel to the line \( y = x\).

Hence, the set \( S \) does not represent a line parallel to \( y = x \). Therefore, Statement-II is false.

Conclusion

The correct option is: Statement-I is true but Statement-II is false.