The statement `p to(q to p)` is equivalent to

Options

A. `p to q`
B. `p to (p vv q)`
C. `p to (p to q)`
D. `p to (p ^^ q)`

Explanation

To determine the equivalence of the statement \( p \to (q \to p) \), we will analyze it step by step using truth tables.

### Step 1: Understand the components
We need to break down the statement \( p \to (q \to p) \). Here, \( p \) and \( q \) are propositions, and \( \to \) represents the conditional (if...then) operator.

### Step 2: Construct the truth table
We will create a truth table that includes all possible truth values for \( p \) and \( q \).

| \( p \) | \( q \) | \( q \to p \) | \( p \to (q \to p) \) |
|---------|---------|---------------|-----------------------|
| T | T | T | T |
| T | F | T | T |
| F | T | F | T |
| F | F | T | T |

### Step 3: Fill in the truth values
1. **Calculate \( q \to p \)**:
- If \( q \) is true and \( p \) is true, then \( q \to p \) is true (T).
- If \( q \) is true and \( p \) is false, then \( q \to p \) is false (F).
- If \( q \) is false and \( p \) is true, then \( q \to p \) is true (T).
- If \( q \) is false and \( p \) is false, then \( q \to p \) is true (T).

2. **Calculate \( p \to (q \to p) \)**:
- If \( p \) is true and \( q \to p \) is true, then \( p \to (q \to p) \) is true (T).
- If \( p \) is true and \( q \to p \) is false, then \( p \to (q \to p) \) is false (F).
- If \( p \) is false and \( q \to p \) is true, then \( p \to (q \to p) \) is true (T).
- If \( p \) is false and \( q \to p \) is false, then \( p \to (q \to p) \) is true (T).

### Step 4: Analyze the final column
From the truth table, we see that \( p \to (q \to p) \) is true in all cases except when \( p \) is true and \( q \) is false. Thus, it is a tautology.

### Step 5: Compare with options
Now we need to compare this with the given options:
1. \( p \to q \)
2. \( p \to (p \lor q) \)
3. \( p \to (p \to q) \)
4. \( p \to (p \land q) \)

We can analyze each option similarly using truth tables or logical reasoning. However, from our analysis, we find that the statement \( p \to (q \to p) \) is equivalent to \( p \to (p \lor q) \).

### Conclusion
The statement \( p \to (q \to p) \) is equivalent to option 2: \( p \to (p \lor q) \).

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