Question

Show that R is an equivalence relation on the set Z of integers given by R = {(a, b) : 2 divides a – b}. Write the equivalence class [0].

Answer 1

To show that R is an equivalence relation on the set Z of integers, we need to prove three properties: reflexive, symmetric, and transitive. The equivalence class [0] consists of all integers that are congruent to 0 modulo 2.

Step-by-step explanation:

To show that R is an equivalence relation on the set Z of integers, we need to prove three properties: reflexive, symmetric, and transitive.

1. Reflexive: For any integer a, a - a = 0, and since 2 divides 0, (a, a) is in R. Therefore, R is reflexive.

2. Symmetric: If (a, b) is in R, then 2 divides a - b. This implies that 2 divides -(a - b), which means 2 divides b - a. Hence, (b, a) is also in R. Therefore, R is symmetric.

3. Transitive: If (a, b) and (b, c) are in R, then 2 divides a - b and 2 divides b - c, which implies 2 divides (a - b) + (b - c) = a - c. Therefore, (a, c) is in R, and R is transitive.

The equivalence class [0] contains all integers that are congruent to 0 modulo 2. This means [0] = {..., -4, -2, 0, 2, 4, ...}.