Assertion and Reason type questions :Consider the following statements, p: Every reflexive relation is a symmetric relation, q: Every anti-symmetric relation is reflexive.Which of the following is/ are true?

Options

A. p alone
B. q alone
C. both p and q
D. neither p nor q

Answer

To solve the problem, we need to analyze the two statements provided:

**Statement p:** Every reflexive relation is a symmetric relation.

**Statement q:** Every anti-symmetric relation is reflexive.

### Step 1: Analyze Statement p
1. **Definition of Reflexive Relation:** A relation R on a set A is called reflexive if for every element a in A, the pair (a, a) is in R.
2. **Definition of Symmetric Relation:** A relation R on a set A is called symmetric if for every pair (a, b) in R, the pair (b, a) is also in R.
3. **Counterexample:** Consider the set A = {1, 2, 3} and the relation R = {(1, 1), (2, 2), (3, 3), (1, 2)}.
- This relation is reflexive because it includes (1, 1), (2, 2), and (3, 3).
- However, it is not symmetric because (1, 2) is in R but (2, 1) is not.
4. **Conclusion for Statement p:** Since we found a counterexample where a reflexive relation is not symmetric, statement p is false.

### Step 2: Analyze Statement q
1. **Definition of Anti-symmetric Relation:** A relation R on a set A is called anti-symmetric if for any pairs (a, b) and (b, a) in R, if a ≠ b, then (a, b) cannot both be in R.
2. **Counterexample:** Consider the set A = {1} and the relation R = ∅ (the empty relation).
- This relation is anti-symmetric because there are no pairs (a, b) to contradict the definition.
- However, it is not reflexive because it does not contain the pair (1, 1).
3. **Conclusion for Statement q:** Since we found a counterexample where an anti-symmetric relation is not reflexive, statement q is also false.

### Final Conclusion
Both statements p and q are false. Therefore, the correct answer is that neither p nor q is true.

### Summary of the Solution
- Statement p is false because a reflexive relation can exist that is not symmetric.
- Statement q is false because an anti-symmetric relation can exist that is not reflexive.
- Thus, the answer is that neither statement is true.