Question

In the set \( W \) of whole numbers, an equivalence relation \( R \) is defined as follows: \( a R b \) iff both \( a \) and \( b \) leave the same remainder when divided by 5. The equivalence class of 1 is given by

• A. \( \{ 2, 7, 12, 17, \dots \} \)
• B. \( \{ 1, 6, 11, 16, \dots \} \)
• C. \( \{ 4, 9, 14, 19, \dots \} \)
• D. \( \{ 0, 5, 10, 15, \dots \} \)

Hint:

To solve problems involving equivalence relations, always focus on the condition specified (in this case, the same remainder when divided by 5). This will help you identify the equivalence class.

Answer 1

The Correct Option is B

The equivalence relation \( a R b \) means that both \( a \) and \( b \) leave the same remainder when divided by 5. We are tasked with finding the equivalence class of 1. The equivalence class of a number is the set of all numbers that give the same remainder when divided by 5. For the number 1, we check all numbers that give a remainder of 1 when divided by 5: \[ 1 \div 5 = 0 \text{ remainder } 1, \quad 6 \div 5 = 1 \text{ remainder } 1, \quad 11 \div 5 = 2 \text{ remainder } 1, \quad 16 \div 5 = 3 \text{ remainder } 1, \dots \] Thus, the equivalence class of 1 is \( \{ 1, 6, 11, 16, \dots \} \). Therefore, the correct answer is option (B).