` p to q` is logically equivalent to

Options

A. `p ^^ ~ q`
B. ` ~ p to ~ q`
C. ` ( p vv ~ q)`
D. none of these

Answer

To determine what `p to q` (or `if p then q`) is logically equivalent to, we can analyze it step by step.

### Step 1: Understanding the Implication
The statement `p to q` can be represented as `p → q`. This means that if `p` is true, then `q` must also be true. The implication is only false when `p` is true and `q` is false.

### Step 2: Constructing the Truth Table
Let's construct the truth table for `p → q`:

| p | q | p → q |
|---|---|-------|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |

From the truth table, we can see that `p → q` is false only when `p` is true and `q` is false.

### Step 3: Identifying Logical Equivalents
We need to find a logical expression that has the same truth values as `p → q`. One known logical equivalence is that `p → q` is equivalent to `¬p ∨ q` (not p or q).

### Step 4: Verifying the Equivalence
Let's create a truth table for `¬p ∨ q` to see if it matches `p → q`.

| p | q | ¬p | ¬p ∨ q |
|---|---|----|--------|
| T | T | F | T |
| T | F | F | F |
| F | T | T | T |
| F | F | T | T |

Now we compare the results of `p → q` and `¬p ∨ q`:

- When `p` is T and `q` is T, both are T.
- When `p` is T and `q` is F, both are F.
- When `p` is F and `q` is T, both are T.
- When `p` is F and `q` is F, both are T.

Since the truth values match, we conclude that `p → q` is logically equivalent to `¬p ∨ q`.

### Conclusion
Thus, `p to q` is logically equivalent to `¬p ∨ q`.

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