Question
Let *, ▢ {, } be such that the Boolean expression (p * q) (p ▢ q) is a tautology. Then :
Options
Solution
Key Concepts and Formulas
- Tautology: A statement that is always true, regardless of the truth values of its components.
- Implication (→): is false only when is true and is false. Otherwise, it is true.
- Logical OR (∨): is true if either or or both are true. It is false only when both and are false.
- Logical AND (∧): is true only when both and are true. Otherwise, it is false.
- Logical NOT (∼): is true when is false, and false when is true.
Step-by-Step Solution
Step 1: Analyze the given expression and the possible operators.
We are given the expression , where * and can be either (AND) or (OR). We need to find the combination of * and that makes the expression a tautology. We will test each option using truth tables.
Step 2: Test option (A): * = , = .
The expression becomes . Let's construct a truth table:
| T | T | F | T | T | T |
| T | F | T | T | T | T |
| F | T | F | F | T | T |
| F | F | T | T | F | F |
Since the last column is not all T, option (A) is incorrect.
Step 3: Test option (B): * = , = .
The expression becomes . Let's construct a truth table:
| T | T | F | F | T | T |
| T | F | T | T | F | F |
| F | T | F | F | F | T |
| F | F | T | F | F | T |
Since the last column is not all T, option (B) is incorrect.
Step 4: Test option (C): * = , = .
The expression becomes . Let's construct a truth table:
| T | T | F | F | T | T |
| T | F | T | T | T | T |
| F | T | F | F | T | T |
| F | F | T | F | F | T |
Since the last column is all T, option (C) is correct.
Step 5: Test option (D): * = , = .
The expression becomes . Let's construct a truth table:
| T | T | F | T | T | T |
| T | F | T | T | F | F |
| F | T | F | F | F | T |
| F | F | T | T | F | F |
Since the last column is not all T, option (D) is incorrect.
Common Mistakes & Tips
- Carefully construct the truth tables, paying close attention to the order of operations and the definitions of the logical operators. A single error in the truth table can lead to the wrong answer.
- Remember that is only false when is true and is false.
- When testing options, start with simpler options first. If a simpler option works, it can save time.
Summary
We tested each possible combination of logical operators for * and in the given Boolean expression to determine which combination results in a tautology. By constructing truth tables for each option, we found that the expression is a tautology, which corresponds to option (C).
Final Answer
The final answer is \boxed{C}, which corresponds to option (C).