Question
If the Boolean expression (p q) (~ p q) is equivalent to p q, where , then the ordered pair is :
Options
Solution
Key Concepts and Formulas
- Truth Tables: A systematic way to evaluate the truth value of a Boolean expression for all possible combinations of truth values of its variables.
- Logical Connectives:
- (AND): is true if and only if both and are true.
- (OR): is true if either or (or both) are true.
- (NOT): is true if is false, and false if is true.
- (Equivalence): Two expressions are equivalent if they have the same truth value for all possible truth values of their variables.
Step-by-Step Solution
Step 1: Understand the given equivalence We are given the Boolean expression , where . We need to find the correct combination of and that makes this equivalence true.
Step 2: Test option (A): , Substitute with and with in the given expression: Now, let's construct the truth table for the left-hand side (LHS) and the right-hand side (RHS) to check for equivalence.
| p | q | |||||
|---|---|---|---|---|---|---|
| T | T | T | T | F | F | T |
| T | F | T | F | T | T | F |
| F | T | T | F | T | T | F |
| F | F | F | F | T | F | F |
Comparing the columns for and , we see that they are not the same. Therefore, option (A) is incorrect. However, the correct answer is given as A. Let's re-examine. The mistake is assuming option A is incorrect because the columns of and are not identical. The problem states the expression is equivalent to , which means the expressions must evaluate to the same truth value. Looking at the truth table above, this is NOT the case.
Step 3: Re-evaluate option (A) and identify the error Let's reconsider option A. It is stated as the correct answer, so we must have an error. We have: Substituting with and with , we get:
The mistake is in the question itself. The given answer is wrong. Let's try option C.
Step 4: Test option (C): , Substitute with and with in the given expression: Let's construct the truth table:
| p | q | |||||
|---|---|---|---|---|---|---|
| T | T | T | T | F | F | T |
| T | F | F | T | F | F | F |
| F | T | F | T | F | F | F |
| F | F | F | F | T | F | F |
Here, and are not equivalent.
Step 5: Test option (B): ,
| p | q | ||||
|---|---|---|---|---|---|
| T | T | T | F | F | T |
| T | F | T | F | F | F |
| F | T | T | F | F | F |
| F | F | F | T | F | F |
Not equivalent.
Step 6: Test option (D): ,
| p | q | ||||
|---|---|---|---|---|---|
| T | T | T | F | F | T |
| T | F | F | T | F | F |
| F | T | F | T | F | F |
| F | F | F | T | F | F |
Not equivalent. The premise of the question seems flawed, as none of the options result in the given equivalence. The closest we can get is for to be always false. If we require the LHS to be always false, then we can see that options B and D work. But none of them result in the LHS being equivalent to .
Common Mistakes & Tips
- Carefully construct the truth tables to avoid errors in evaluating the Boolean expressions.
- Double-check the logical connectives and their truth values.
- Remember that equivalence means the truth values must be the same for all possible combinations of the variables.
Summary The given answer and the premise of the question seem incorrect. None of the options satisfy the given equivalence . The closest one can get is the expression evaluating to always false.
Final Answer The question seems flawed. None of the options are correct. There might be a typo in the question or the given correct answer.