GATE CS 2015 Set 1 — Question 40
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Digital Logic → Boolean Algebra & Logic Gates → Boolean Laws & Theorems
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Question
The binary operator is defined by the following truth table.
Which one of the following is true about the binary operator ?
| p | q | p q |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
Correct answer
(A) Both commutative and associative
Solution
The given binary operator is defined by the truth table:
This truth table corresponds to the XOR (exclusive OR) operation.
Since , and similarly for other pairs (e.g., , ), the operation is commutative.
For example, let :
The results are equal.
Let :
The results are equal.
Since XOR is both commutative and associative, option A is correct.The final answer is
| p | q | p q |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
1.Commutativity: An operation is commutative if .
From the table:Since , and similarly for other pairs (e.g., , ), the operation is commutative.
2.Associativity: An operation is associative if .
Let's test with all possible combinations of . The XOR operation is known to be associative.For example, let :
The results are equal.
Let :
The results are equal.
Since XOR is both commutative and associative, option A is correct.The final answer is
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