The statement `(~(phArr q))^^p` is equivalent to

Options

A. `p^^ q`
B. `qhArr p`
C. `p^^~q`
D. `~p^^q`

Answer

To determine which statement is equivalent to `(~(P ↔ Q)) ∧ P`, we will create a truth table for the expression and analyze the results step by step.

### Step 1: Define the Variables
Let:
- P: A proposition
- Q: Another proposition

### Step 2: Create the Truth Table
We will create a truth table that includes the columns for P, Q, P ↔ Q, ~(P ↔ Q), and finally ~(P ↔ Q) ∧ P.

| P | Q | P ↔ Q | ~(P ↔ Q) | ~(P ↔ Q) ∧ P |
|-------|-------|-------|----------|---------------|
| T | T | T | F | F |
| T | F | F | T | T |
| F | T | F | T | F |
| F | F | T | F | F |

### Step 3: Fill in the Truth Table
1. **P ↔ Q**: This is true when both P and Q are the same (both true or both false).
- T, T → T
- T, F → F
- F, T → F
- F, F → T

2. **~(P ↔ Q)**: This is the negation of the previous column.
- T → F
- F → T
- F → T
- T → F

3. **~(P ↔ Q) ∧ P**: This is true when both ~(P ↔ Q) is true and P is true.
- F ∧ T → F
- T ∧ T → T
- T ∧ F → F
- F ∧ F → F

### Step 4: Analyze the Results
From the truth table, we see that the expression `(~(P ↔ Q)) ∧ P` evaluates to true only when P is true and Q is false.

### Conclusion
The equivalent statement to `(~(P ↔ Q)) ∧ P` is the one that matches the truth values in the last column of the truth table.