If a relation R on the set of integers Z is defined

Solution

We are given:

• The set Z (set of all integers)
• A relation R is defined by:
  a R b if and only if (a - b) is a rational number

Now let's check the three properties one by one: Reflexive: We check if a R a is true for all integers a. a - a = 0, and 0 is a rational number. So a R a is true for all a ∈ Z. → The relation is reflexive. Symmetric: Suppose a R b is true. Then a - b is rational. Now b - a = -(a - b), which is also rational (since the negative of a rational number is rational).
So b R a is also true. → The relation is symmetric. Transitive:
Suppose a R b and b R c are true.
Then a - b and b - c are both rational.
Add them: (a - b) + (b - c) = a - c, which is also rational.
So a R c is true.
→ The relation is transitive. The relation R is reflexive, symmetric, and transitive. Hence, it is an equivalence relation . Final Answer: (A) Reflexive, symmetric, transitive