Question

A relation \( R \) defined on set \( A = \{x : x \in \mathbb{Z} \text{ and } 0 \leq x \leq 10\} \) as \( R = \{(x, y) : x = y\) is given to be an equivalence relation. The number of equivalence classes is:

• A. 1
• B. 2
• C. 10
• D. 11
• E.

Hint:

For an equality relation \( R = \{(x, y) : x = y\} \) on a finite set, the number of equivalence classes is equal to the number of elements in the set.

Answer 1

The Correct Option is D

The relation \( R = \{(x, y) : x = y\} \) is the equality relation, meaning \( x \) is equivalent to \( y \) only if \( x = y \). The given set is: \[ A = \{0, 1, 2, \dots, 10\}. \] 1. Equivalence Relation: - Reflexive: For all \( x \in A \), \( (x, x) \in R \). True, since \( x = x \). - Symmetric: If \( (x, y) \in R \), then \( (y, x) \in R \). True, since \( x = y \) implies \( y = x \). - Transitive: If \( (x, y) \in R \) and \( (y, z) \in R \), then \( (x, z) \in R \). True, since \( x = y \) and \( y = z \) imply \( x = z \). Therefore, \( R \) is an equivalence relation. 2. Equivalence Classes: Each element of \( A \) forms its own equivalence class because \( x = y \) only holds for a single \( y \). Hence, the equivalence classes are: \[ \{0\}, \{1\}, \{2\}, \dots, \{10\}. \] The total number of equivalence classes is equal to the number of elements in \( A \), which is: \[ 11. \] Hence, the correct answer is (D) 11.