Which of the following option is incorrect?

Options

A. `~(phArr ~q) -=phArr q`
B. `~(p hArr q)-=(prarr~q)^^(~qrarrp)`
C. `~(phArr q)-=~p rArr q`
D. `~(prArrq)-=p^^~q`

Answer

To determine which of the given options is incorrect, we will analyze each option by constructing truth tables for the logical expressions provided in both the Left-Hand Side (LHS) and Right-Hand Side (RHS).

### Step-by-Step Solution:

#### Option 1:
1. **LHS**: \( P \equiv \neg Q \)
- Construct the truth table for \( P \) and \( Q \):
- \( P: \) T, T, F, F
- \( Q: \) T, F, T, F
- \( \neg Q: \) F, T, F, T
- Now calculate \( P \equiv \neg Q \):
- T ≡ F → F
- T ≡ T → T
- F ≡ F → T
- F ≡ T → F
- Result: F, T, T, F

2. **RHS**: \( \neg P \equiv \neg Q \)
- Calculate \( \neg P \):
- \( \neg P: \) F, F, T, T
- Now calculate \( \neg P \equiv \neg Q \):
- F ≡ F → T
- F ≡ T → F
- T ≡ F → F
- T ≡ T → T
- Result: T, F, F, T

3. **Comparison**: LHS = F, T, T, F and RHS = T, F, F, T
- They are not equal, thus this option is incorrect.

#### Option 2:
1. **LHS**: \( \neg P \equiv Q \)
- Calculate \( \neg P \):
- \( \neg P: \) F, F, T, T
- Now calculate \( \neg P \equiv Q \):
- F ≡ T → F
- F ≡ F → T
- T ≡ T → T
- T ≡ F → F
- Result: F, T, T, F

2. **RHS**: \( P \Rightarrow \neg Q \)
- Calculate \( \neg Q \):
- \( \neg Q: \) F, T, F, T
- Now calculate \( P \Rightarrow \neg Q \):
- T → F → F
- T → T → T
- F → F → T
- F → T → T
- Result: F, T, T, T

3. **Comparison**: LHS = F, T, T, F and RHS = F, T, T, T
- They are not equal, thus this option is incorrect.

#### Option 3:
1. **LHS**: \( \neg P \Rightarrow Q \)
- Calculate \( \neg P \):
- \( \neg P: \) F, F, T, T
- Now calculate \( \neg P \Rightarrow Q \):
- F → T → T
- F → F → T
- T → T → T
- T → F → F
- Result: T, T, T, F

2. **RHS**: \( \neg P \equiv Q \)
- From previous calculations, we have:
- Result: F, T, T, F

3. **Comparison**: LHS = T, T, T, F and RHS = F, T, T, F
- They are not equal, thus this option is incorrect.

#### Option 4:
1. **LHS**: \( P \land \neg Q \)
- Calculate \( \neg Q \):
- \( \neg Q: \) F, T, F, T
- Now calculate \( P \land \neg Q \):
- T ∧ F → F
- T ∧ T → T
- F ∧ F → F
- F ∧ T → F
- Result: F, T, F, F

2. **RHS**: \( P \land \neg Q \)
- From previous calculations, we have:
- Result: F, T, F, F

3. **Comparison**: LHS = F, T, F, F and RHS = F, T, F, F
- They are equal, thus this option is correct.

### Conclusion:
The incorrect options are **Option 1**, **Option 2**, and **Option 3**.