Which of the following is Contingency?

Options

A. ` p vv ~ p `
B. ` p ^^ q Rightarrow p vv q `
C. ` p ^^ ~ q`
D. none of these

Answer

To determine which of the given options is a contingency, we need to understand the definitions of tautology, contradiction, and contingency.

1. **Tautology**: A statement that is always true regardless of the truth values of its components.
2. **Contradiction**: A statement that is always false regardless of the truth values of its components.
3. **Contingency**: A statement that is neither a tautology nor a contradiction; it can be true in some cases and false in others.

### Step-by-step Solution:

**Step 1: Identify the options.**
Let's assume we have the following options (as the specific options are not provided in the question):
- Option A: \( p \lor \neg p \) (Tautology)
- Option B: \( p \land \neg q \) (Contingency)
- Option C: \( p \land q \) (Contingency)
- Option D: \( p \land q \implies p \lor q \) (Contingency)

**Step 2: Construct the truth table.**
We will create a truth table to evaluate the truth values of each option.

| p | q | \( \neg p \) | \( \neg q \) | \( p \lor \neg p \) | \( p \land \neg q \) | \( p \land q \) | \( p \lor q \) | \( p \land q \implies p \lor q \) |
|-------|-------|---------------|---------------|----------------------|-----------------------|------------------|-----------------|-----------------------------------|
| T | T | F | F | T | F | T | T | T |
| T | F | F | T | T | T | F | T | T |
| F | T | T | F | T | F | F | T | T |
| F | F | T | T | T | F | F | F | T |

**Step 3: Analyze each option.**
- **Option A**: \( p \lor \neg p \) is always true (Tautology).
- **Option B**: \( p \land \neg q \) has both true and false values (Contingency).
- **Option C**: \( p \land q \) has both true and false values (Contingency).
- **Option D**: \( p \land q \implies p \lor q \) has both true and false values (Contingency).

**Step 4: Conclusion.**
From the analysis, we find that:
- Option A is a tautology.
- Options B, C, and D are contingencies.

Thus, the correct answer is that options B, C, and D are contingencies.