Which of the following is true :

Options

A. `p ^^ q=p vv q`
B. `(p ^^ q vv r) ltimplies (p vv q) ^^ r`
C. `~| (p ^^ q) ltimplies ~| p vv ~| q`
D. `~| p = ~| q`

Answer

To determine which of the given statements is true, we will analyze each option step by step.

### Step 1: Analyze Option A
**Statement:** \( P \land Q \equiv P \lor Q \)

- **Explanation:**
- Here, \( \land \) represents conjunction (AND) and \( \lor \) represents disjunction (OR).
- The conjunction \( P \land Q \) is true only when both \( P \) and \( Q \) are true.
- The disjunction \( P \lor Q \) is true when at least one of \( P \) or \( Q \) is true.
- Since the conditions for truth are different for conjunction and disjunction, this statement is not true.

### Conclusion for Option A:
- **Result:** False

### Step 2: Analyze Option B
**Statement:** \( P \land Q \land R \equiv P \lor Q \land R \)

- **Explanation:**
- According to the associative law of conjunction, \( P \land (Q \land R) \equiv (P \land Q) \land R \).
- The left side \( P \land Q \land R \) is true only when all three variables are true.
- The right side \( P \lor Q \land R \) is true if either \( P \) is true or both \( Q \) and \( R \) are true.
- Since the conditions for truth are different, this statement is also not true.

### Conclusion for Option B:
- **Result:** False

### Step 3: Analyze Option C
**Statement:** \( \neg (P \land Q) \equiv \neg P \lor \neg Q \)

- **Explanation:**
- This statement is an application of De Morgan's Laws, which state that the negation of a conjunction is equivalent to the disjunction of the negations.
- According to De Morgan's Laws, \( \neg (P \land Q) \) is indeed equivalent to \( \neg P \lor \neg Q \).
- Therefore, this statement is true.

### Conclusion for Option C:
- **Result:** True

### Step 4: Analyze Option D
**Statement:** \( \neg P \equiv \neg Q \)

- **Explanation:**
- This statement claims that the negation of \( P \) is equivalent to the negation of \( Q \).
- This is not always true; it depends on the truth values of \( P \) and \( Q \).
- For example, if \( P \) is true and \( Q \) is false, then \( \neg P \) is false and \( \neg Q \) is true, which shows they are not equivalent.

### Conclusion for Option D:
- **Result:** False

### Final Conclusion:
- The only true statement among the options is **Option C**: \( \neg (P \land Q) \equiv \neg P \lor \neg Q \).

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