The statement `A to (B to A)` is equivalent to :

Options

A. `A to (A wedge B)`
B. `A to (A to B)`
C. `A to (A harr B)`
D. `A to (A vee B)`

Answer

To solve the problem of determining what the statement \( A \to (B \to A) \) is equivalent to, we will create a truth table for the expression and analyze each option provided.

### Step 1: Understand the expression
The expression \( A \to (B \to A) \) can be broken down. The inner expression \( B \to A \) means "if B is true, then A is true." The outer expression \( A \to (B \to A) \) means "if A is true, then (if B is true, A is true)."

### Step 2: Create the truth table
We will create a truth table for \( A \) and \( B \), and then calculate \( B \to A \) and \( A \to (B \to A) \).

| A | B | B → A | A → (B → A) |
|-------|-------|-------|-------------|
| True | True | True | True |
| True | False | True | True |
| False | True | False | True |
| False | False | True | True |

### Explanation of the truth table:
- **Column 1 and 2**: All combinations of truth values for \( A \) and \( B \).
- **Column 3**: \( B \to A \) is true unless \( B \) is true and \( A \) is false.
- **Column 4**: \( A \to (B \to A) \) is true in all cases. This is because if \( A \) is false, the implication \( A \to (B \to A) \) is true regardless of the truth value of \( B \).

### Step 3: Analyze the results
From the truth table, we can see that \( A \to (B \to A) \) is always true. This means that the expression is equivalent to a tautology.

### Conclusion
The statement \( A \to (B \to A) \) is equivalent to a tautology, which can be represented as \( \text{True} \).