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

Options

A. p
B. q
C. `~ p vee q `
D. `p vee ~ q `

Answer

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

### Step 1: Create a Truth Table
We need to evaluate the truth values of \( p \), \( q \), \( p \rightarrow q \), and \( (p \rightarrow q) \rightarrow p \).

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

### Step 2: Fill in the Columns
1. **Column for \( p \rightarrow q \)**:
- \( p \rightarrow q \) is false only when \( p \) is true and \( q \) is false. In all other cases, it is true.

2. **Column for \( (p \rightarrow q) \rightarrow p \)**:
- This expression is true unless \( (p \rightarrow q) \) is true and \( p \) is false. We can analyze this based on the previous column:
- When \( p \) is true, \( (p \rightarrow q) \rightarrow p \) is true regardless of \( q \).
- When \( p \) is false, \( (p \rightarrow q) \) is true, but \( p \) is false, making the whole expression false.

### Step 3: Analyze the Results
From the truth table, we see that the final column for \( (p \rightarrow q) \rightarrow p \) yields the results:
- T, T, F, F

### Step 4: Determine the Equivalent Statement
Now we need to find which of the options matches the truth values of \( (p \rightarrow q) \rightarrow p \):
- The only option that matches the truth values of T, T, F, F is \( p \).

### Conclusion
Thus, the statement \( (p \rightarrow q) \rightarrow p \) is equivalent to \( p \).

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