Statement-1: On the set Z of all odd integers relation R defined by
`(a, b) in R iff a-b` is even for all `a, b in Z` is an equivalence relation.
Statement-2: If a relation R on a set A is symmetric and transitive, then it is reflexive and hence an equivalence relation, because
`(a, b) in Rimplies(b, a)in R" [By symmetry]"`
`(a, b)in R and (b, a) in Rimplies (a,a)in R " [By transitivity]"`

Options

A. Statement-1 is True, Statement-2 is true, Statement-2 is a correct explanation for statement-1.
B. Statement-1 is True, Statement-2 is true, Statement-2 is not a correct explanation for Statement-1.
C. Statement-1 is True, Statement-2 is False
D. Statement-1 is False, Statement-2 is True.

Explanation

To solve the given problem, we will analyze both statements step by step.

### Step 1: Analyze Statement 1
**Statement 1:** On the set Z of all odd integers, relation R defined by (a, b) in R iff a - b is even for all a, b in Z is an equivalence relation.

**Solution:**
1. **Reflexivity:** For any odd integer a, a - a = 0, which is even. Thus, (a, a) is in R for all a in Z. Therefore, R is reflexive.

2. **Symmetry:** If (a, b) is in R, then a - b is even. This implies that b - a = -(a - b) is also even. Hence, (b, a) is in R. Therefore, R is symmetric.

3. **Transitivity:** If (a, b) is in R and (b, c) is in R, then a - b is even and b - c is even. Adding these gives (a - b) + (b - c) = a - c, which is even. Thus, (a, c) is in R. Therefore, R is transitive.

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

### Conclusion for Statement 1:
**Statement 1 is true.**

### Step 2: Analyze Statement 2
**Statement 2:** If a relation R on a set A is symmetric and transitive, then it is reflexive and hence an equivalence relation.

**Solution:**
1. **Counterexample:** Consider a relation R on set A = {1, 2} defined as R = {(1, 2), (2, 1)}.
- This relation is symmetric because (1, 2) implies (2, 1).
- It is transitive because there are no pairs (a, b) and (b, c) such that a ≠ c.
- However, it is not reflexive because (1, 1) and (2, 2) are not in R.

Since we found a counterexample where R is symmetric and transitive but not reflexive, Statement 2 is false.

### Conclusion for Statement 2:
**Statement 2 is false.**

### Final Conclusion:
- Statement 1 is true.
- Statement 2 is false.

### Correct Option:
The correct option is that Statement 1 is true and Statement 2 is false.

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