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 q and q
D. neither p nor q

Answer

To solve the assertion and reason type question regarding the statements about reflexive, symmetric, and anti-symmetric relations, we will analyze each statement step by step.

### Step 1: Analyze Statement P
**Statement P**: Every reflexive relation is a symmetric relation.

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. This means every element is related to itself.

2. **Definition of Symmetric Relation**: A relation R on a set A is called symmetric if for any elements a and b in A, if (a, b) is in R, then (b, a) must also be in R.

3. **Analysis**: If R is reflexive, it contains pairs of the form (a, a). For any such pair, since a = a, the condition for symmetry is satisfied (i.e., if (a, a) is in R, then (a, a) is also in R). Therefore, every reflexive relation is indeed symmetric.

**Conclusion for Statement P**: True.

### Step 2: Analyze Statement Q
**Statement Q**: Every anti-symmetric relation is reflexive.

1. **Definition of Anti-symmetric Relation**: A relation R on a set A is called anti-symmetric if for any elements a and b in A, if (a, b) is in R and (b, a) is also in R, then a must equal b.

2. **Analysis**: For a relation to be anti-symmetric, it does not necessarily mean that (a, a) must be included in R for every a in A. The anti-symmetric condition only applies when both (a, b) and (b, a) are present in R. It is possible to have an anti-symmetric relation that does not include any pairs of the form (a, a) for some elements. Therefore, an anti-symmetric relation does not guarantee reflexivity.

**Conclusion for Statement Q**: False.

### Final Conclusion
- Statement P is true.
- Statement Q is false.

Thus, the correct answer is that only statement P is true.