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

Options

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

Explanation

To determine the equivalence of the statement \( p \to (q \to p) \), we will construct a truth table and analyze the results step by step.

### Step 1: Define the Statements
We have two statements:
- \( p \): A proposition
- \( q \): Another proposition

The statement we need to analyze is \( p \to (q \to p) \).

### Step 2: Understand Implication
Recall that the implication \( a \to b \) is equivalent to \( \neg a \lor b \). This means:
- \( p \to (q \to p) \) can be rewritten as \( \neg p \lor (q \to p) \).
- Further, \( q \to p \) is equivalent to \( \neg q \lor p \).

### Step 3: Construct the Truth Table
We will create a truth table for \( p \), \( q \), \( q \to p \), and \( p \to (q \to p) \).

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

### Step 4: Analyze the Truth Table
- For \( p = T \) and \( q = T \): \( q \to p \) is T, hence \( p \to (q \to p) \) is T.
- For \( p = T \) and \( q = F \): \( q \to p \) is T, hence \( p \to (q \to p) \) is T.
- For \( p = F \) and \( q = T \): \( q \to p \) is F, but \( p \to (q \to p) \) is T (since \( p \) is F).
- For \( p = F \) and \( q = F \): \( q \to p \) is T, hence \( p \to (q \to p) \) is T.

### Step 5: Conclusion
From the truth table, we can see that \( p \to (q \to p) \) is always true regardless of the truth values of \( p \) and \( q \). Therefore, the statement \( p \to (q \to p) \) is a tautology.