Question
Which of the following is equivalent to the Boolean expression p q ?
Options
Solution
Key Concepts and Formulas
- Logical AND (): The statement is true if and only if both and are true.
- Logical NOT (): The statement is true if is false, and false if is true.
- Logical Implication (): The statement is equivalent to . It is only false when is true and is false. We can also write .
Step-by-Step Solution
Step 1: Construct the truth table for . We need to find the truth values for , , and then .
| T | T | F | F |
| T | F | T | T |
| F | T | F | F |
| F | F | T | F |
Step 2: Construct the truth table for . First, we find , then we negate it.
| T | T | T | F |
| T | F | T | F |
| F | T | F | T |
| F | F | T | F |
Step 3: Construct the truth table for . First, we find and , then we find .
| T | T | F | F | T |
| T | F | F | T | T |
| F | T | T | F | F |
| F | F | T | T | T |
Step 4: Construct the truth table for . First, we find , then , and then .
| T | T | F | F | T |
| T | F | T | T | F |
| F | T | F | T | F |
| F | F | T | T | F |
Step 5: Construct the truth table for . First, we find , then we negate it.
| T | T | T | F |
| T | F | F | T |
| F | T | T | F |
| F | F | T | F |
Step 6: Compare the truth tables. We are looking for the option that has the same truth table as . Comparing the truth tables from Steps 1-5, we see that in Step 5 has the same truth table as in Step 1.
| T | T | F | F | T | T | F |
| T | F | T | F | T | F | T |
| F | T | F | T | F | F | F |
| F | F | F | F | T | F | F |
We are given that the correct answer is (A), which corresponds to . However, the truth table for does not match the truth table for . There appears to be an error in the provided answer.
We see that the truth table for is identical to the truth table for .
Common Mistakes & Tips
- Remember the truth tables for basic logical operations (AND, OR, NOT, Implication).
- When negating an implication, use the equivalence .
- Be careful when constructing truth tables, especially with multiple negations or implications.
Summary
We constructed the truth tables for each of the given options and compared them to the truth table for . By comparing the truth tables, we find that . However, the question states that the correct answer is option (A), . There appears to be an error in the problem statement, as the correct answer should be option (D), .
Final Answer
The final answer is \boxed{D}, which corresponds to option (D).