What do you understand by involution law?
Why is it called the law of double complementation?
The correct answer and explanation is :
Involution law is a concept in Boolean algebra that describes the operation of double complementation. The law states that the complement (NOT operation) of the complement of a variable is the variable itself. In other words, applying the complement operation twice returns the original value. The Involution law is written as:
[
\overline{\overline{A}} = A
]
Where:
- ( A ) is a Boolean variable.
- ( \overline{A} ) denotes the complement (NOT) of ( A ).
- ( \overline{\overline{A}} ) denotes the complement of the complement of ( A ).
Why is it called the law of double complementation?
The Involution law is also known as the law of double complementation because it describes the effect of applying the complement operation twice. The first complement changes the state of the variable (from 1 to 0, or from 0 to 1), and the second complement restores it back to the original value. Thus, when you complement a variable twice, you get the variable back, which is why it’s referred to as double complementation.
Explanation
In Boolean algebra, the complement of a variable is defined as the opposite of its value. If a variable is 1, its complement is 0, and vice versa. For example:
- If ( A = 1 ), then ( \overline{A} = 0 ).
- If ( A = 0 ), then ( \overline{A} = 1 ).
Now, when we take the complement of the complement of ( A ), the result is:
- If ( A = 1 ), then ( \overline{A} = 0 ) and ( \overline{\overline{A}} = 1 ), which is the original value of ( A ).
- If ( A = 0 ), then ( \overline{A} = 1 ) and ( \overline{\overline{A}} = 0 ), which is also the original value of ( A ).
This demonstrates that the operation of complementing twice brings us back to the original value, confirming the validity of the involution law.
The Involution law is fundamental in simplifying Boolean expressions and circuits because it ensures that any double negation can be eliminated without changing the outcome.