Which of the following is true?

Options

A. `p^^~p-=t`
B. `pvv~p-=f`
C. `pto q-=q to p`
D. `ptoq-=~q to ~p`

Explanation

To solve the question "Which of the following is true?" we need to evaluate the four given conditions step by step. Let's denote the conditions as A, B, C, and D.

### Step 1: Evaluate Condition A
**Condition A:** \( p \land \neg p \)

1. **Truth Values for p:**
- True (T)
- False (F)

2. **Negation of p (\( \neg p \)):**
- If \( p \) is True, \( \neg p \) is False.
- If \( p \) is False, \( \neg p \) is True.

3. **Truth Table for \( p \land \neg p \):**
- For \( p = T \), \( \neg p = F \) → \( T \land F = F \)
- For \( p = F \), \( \neg p = T \) → \( F \land T = F \)

4. **Conclusion for Condition A:**
- \( p \land \neg p \) is always False.

### Step 2: Evaluate Condition B
**Condition B:** \( p \lor \neg p \)

1. **Truth Values for p:**
- True (T)
- False (F)

2. **Negation of p (\( \neg p \)):**
- If \( p \) is True, \( \neg p \) is False.
- If \( p \) is False, \( \neg p \) is True.

3. **Truth Table for \( p \lor \neg p \):**
- For \( p = T \), \( \neg p = F \) → \( T \lor F = T \)
- For \( p = F \), \( \neg p = T \) → \( F \lor T = T \)

4. **Conclusion for Condition B:**
- \( p \lor \neg p \) is always True.

### Step 3: Evaluate Condition C
**Condition C:** \( p \Rightarrow q \) and \( q \Rightarrow p \)

1. **Truth Table for \( p \Rightarrow q \):**
- \( T \Rightarrow T = T \)
- \( T \Rightarrow F = F \)
- \( F \Rightarrow T = T \)
- \( F \Rightarrow F = T \)

2. **Truth Table for \( q \Rightarrow p \):**
- \( T \Rightarrow T = T \)
- \( T \Rightarrow F = F \)
- \( F \Rightarrow T = T \)
- \( F \Rightarrow F = T \)

3. **Combined Truth Table for \( p \Rightarrow q \) and \( q \Rightarrow p \):**
- The only case where both are False is when \( p = T \) and \( q = F \).

4. **Conclusion for Condition C:**
- \( p \Rightarrow q \) and \( q \Rightarrow p \) are not always equal.

### Step 4: Evaluate Condition D
**Condition D:** \( p \Rightarrow q \) and \( \neg p \Rightarrow \neg q \)

1. **Truth Table for \( p \Rightarrow q \):**
- As calculated above.

2. **Truth Table for \( \neg p \Rightarrow \neg q \):**
- If \( p = T \), \( q = T \) → \( F \Rightarrow F = T \)
- If \( p = T \), \( q = F \) → \( F \Rightarrow T = T \)
- If \( p = F \), \( q = T \) → \( T \Rightarrow F = F \)
- If \( p = F \), \( q = F \) → \( T \Rightarrow T = T \)

3. **Conclusion for Condition D:**
- Both \( p \Rightarrow q \) and \( \neg p \Rightarrow \neg q \) are equivalent.

### Final Conclusion
After evaluating all four conditions, we find that:

- Condition A is False.
- Condition B is True.
- Condition C is not always equal.
- Condition D is True.

Thus, the true condition is **Condition B** and **Condition D**.