If A is a non-empty set, then which of the following is `{:(f,a,l,s,e,?):}`
p: There is at least one reflexive relation on A
q: There is at least one symmetric relation on A

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 statements regarding the properties of relations on a non-empty set \( A \).

### Step 1: Understand the definitions of reflexive and symmetric relations.

- **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 must be related to itself.

- **Symmetric Relation**: A relation \( R \) on a set \( A \) is called symmetric if for any two elements \( a, b \in A \), whenever \( (a, b) \) is in \( R \), then \( (b, a) \) must also be in \( R \).

### Step 2: Analyze statement P.

- Given that \( A \) is a non-empty set, let's consider a simple case where \( A = \{ a \} \).
- For this set, the only possible relation that can be formed is \( R = \{ (a, a) \} \).
- Since \( (a, a) \) is in \( R \), this relation is reflexive.
- Therefore, statement P is **true**.

### Step 3: Analyze statement Q.

- Again, consider the same set \( A = \{ a \} \).
- The only relation we can have is \( R = \{ (a, a) \} \).
- In this case, since there are no other elements to relate, the condition for symmetry is trivially satisfied because there are no pairs \( (a, b) \) where \( a \neq b \).
- Hence, the relation \( R \) is symmetric.
- Therefore, statement Q is also **true**.

### Step 4: Conclusion.

- Since both statements P and Q are true, we need to determine which of the statements is false.
- The options given are:
- A: P alone
- B: Q alone
- C: Both P and Q
- D: Neither P nor Q
- Since neither of the statements is false, the correct answer is **D: Neither P nor Q**.

### Final Answer:
The correct option is **D: Neither P nor Q**.

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