Question

Let Z be the set of all integers and let R be a relation on Z defined by a R b <-> (a-b) is divisible by 3 . Then, R is?

A. reflexive and symmetric but not transitive
B. reflextive and transitive but not symmetric
C. syymetric and transitive but not reflextive
D. an equivalence relation

Answer 1

The relation R on the set of integers Z, defined by (a-b) divisible by 3, is reflexive, symmetric, and transitive. Hence, it is an equivalence relation.

Step-by-step explanation:

When analyzing the relation R on the set of integers Z, where a R b if and only if (a-b) is divisible by 3, we can determine its properties of reflexivity, symmetry, and transitivity.

The relation R is an equivalence relation. An equivalence relation must be reflexive, symmetric, and transitive.

Reflexive: For any integer a, a-a = 0, and 0 is divisible by 3. Therefore, every integer is related to itself and R is reflexive.

Symmetric: For any integers a and b, if a-b is divisible by 3, then b-a is also divisible by 3. Therefore, R is symmetric.

Transitive: For any integers a, b, and c, if a-b is divisible by 3 and b-c is divisible by 3, then a-c is also divisible by 3. Therefore, R is transitive.

Since the relation R is reflexive, symmetric, and transitive, it is an equivalence relation.

Therefore answer is D. an equivalence relation.