Which of these is a tautology:

Options

A. `p implies q`
B. `~| p ^^ q`
C. `(p ^^ q) implies q`
D. `p ^^ q`

Answer

To determine which of the given statements is a tautology, we will follow these steps:

### Step 1: Understand Tautology
A tautology is a logical statement that is always true regardless of the truth values of its components.

### Step 2: Analyze the Options
Assuming we have four logical statements (let's denote them as A, B, C, and D), we need to analyze each statement to see if it can be true under all circumstances.

### Step 3: Create Truth Tables
For each statement, we will create a truth table. A truth table lists all possible truth values of the components and the resulting truth value of the statement.

1. **For Statement A**:
- Identify the variables involved.
- List all combinations of truth values (True/False).
- Evaluate the statement based on these combinations.

2. **For Statement B**:
- Repeat the process of creating a truth table.

3. **For Statement C**:
- Again, create a truth table.

4. **For Statement D**:
- Create a truth table for this statement as well.

### Step 4: Evaluate the Results
After constructing the truth tables for all statements, we will check the final column of each table:

- If the final column for a statement contains all True values, then that statement is a tautology.

### Step 5: Identify the Tautology
Once we have evaluated all statements, we can conclude which one is a tautology based on our findings.

### Example of a Tautology
For instance, the statement \( P \lor \neg P \) (P or not P) is a classic example of a tautology because it is always true regardless of whether P is true or false.

### Conclusion
After analyzing all statements, we will identify the one that is always true as the tautology.

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