Which of the following is wrong?

Options

A. ` p to q` is logically equivation to ` ~p vv q`
B. if The truth values of p,q,r are T,F,T respectively, then the truth value of ` ( p vv q) ^^ ( q vv r) ` is T
C. ` ~ ( p vv q vv r) cong -~ p ^^ ~ q ^^ ~ r`
D. the truth value of ` p ^^ ~ ( p vv q)` is always T

Explanation

To determine which of the statements is wrong, we will analyze each statement one by one using logical reasoning and truth tables.

### Step 1: Analyze Statement 1
**Statement 1:** \( p \implies q \) is logically equivalent to \( \neg p \lor q \).

**Truth Table:**
| p | q | \( \neg p \) | \( p \implies q \) | \( \neg p \lor q \) |
|-------|-------|---------------|---------------------|----------------------|
| T | T | F | T | T |
| T | F | F | F | F |
| F | T | T | T | T |
| F | F | T | T | T |

From the truth table, we see that the columns for \( p \implies q \) and \( \neg p \lor q \) are identical. Therefore, **Statement 1 is correct**.

### Step 2: Analyze Statement 2
**Statement 2:** If the truth values of \( p, q, r \) are true, false, and true respectively, then the truth value of \( p \lor q \) and \( p \lor r \) is true.

**Evaluation:**
- \( p = T \)
- \( q = F \)
- \( r = T \)

Calculating:
- \( p \lor q = T \lor F = T \)
- \( p \lor r = T \lor T = T \)

Both expressions are true. Therefore, **Statement 2 is correct**.

### Step 3: Analyze Statement 3
**Statement 3:** \( \neg (p \lor q \lor r) \) is equivalent to \( \neg p \land \neg q \land \neg r \).

**Using De Morgan's Law:**
According to De Morgan's Law, \( \neg (A \lor B) = \neg A \land \neg B \).
So, \( \neg (p \lor q \lor r) = \neg p \land \neg q \land \neg r \).

Thus, **Statement 3 is correct**.

### Step 4: Analyze Statement 4
**Statement 4:** The truth value of \( p \land (\neg p \lor q) \) is always true.

**Truth Table:**
| p | q | \( \neg p \) | \( \neg p \lor q \) | \( p \land (\neg p \lor q) \) |
|-------|-------|---------------|----------------------|---------------------------------|
| T | T | F | T | T |
| T | F | F | F | F |
| F | T | T | T | F |
| F | F | T | T | F |

From the truth table, we see that \( p \land (\neg p \lor q) \) is not always true; it is false when \( p \) is true and \( q \) is false, and also when \( p \) is false regardless of \( q \). Therefore, **Statement 4 is incorrect**.

### Conclusion
The statement that is wrong is **Statement 4**.

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