The idempotent law of Boolean algebra says:a)X.X.X.X.X…. = Xb)X + X + X…. = Xc)A + BC = (A + B)(A + C)d)Both (a) and (b)Correct answer is option 'D'. Can you explain this answer?

Question

Answer

Idempotent law of Boolean algebra:

The idempotent law of Boolean algebra is a fundamental law that states that any Boolean variable remains unchanged when multiplied by itself any number of times. It is a very useful law and is used in many applications of Boolean algebra. The idempotent law is also very simple to understand and apply.

The idempotent law can be expressed as follows:

X.X.X.X.X…. = X
X X X…. = X

Explanation:

The idempotent law of Boolean algebra states that any Boolean variable remains unchanged when multiplied by itself any number of times. This means that if we have a Boolean variable X, and we multiply it by itself any number of times, the result will always be X. This is because the value of X does not change, regardless of how many times it is multiplied by itself.

For example, if we have X = 1, then X.X.X.X.X…. = 1. Similarly, if X = 0, then X.X.X.X.X…. = 0. This is because the value of X does not change, regardless of how many times it is multiplied by itself.

Similarly, if we have X X X…. = X, then it means that any number of X's multiplied together will always result in X. This is again because the value of X does not change, regardless of how many times it is multiplied by itself.

Applications:

The idempotent law of Boolean algebra is used in many applications of Boolean algebra, such as digital logic design, computer science, and electronic engineering. It is used to simplify Boolean expressions and to reduce the complexity of digital circuits.

For example, if we have a Boolean expression A.B.C.A.B.C, we can simplify it using the idempotent law as follows:

A.B.C.A.B.C = A.B.C

This is because we can remove the repeated terms A.B.C and simplify the expression using the idempotent law.

Conclusion:

The idempotent law of Boolean algebra is a fundamental law that states that any Boolean variable remains unchanged when multiplied by itself any number of times. It is a simple and useful law that is used in many applications of Boolean algebra.

Name	AND Form	OR Form
Identity law	1.A = A	0 + A = A
Null Law	0.A = 0	1 + A = 1
Idempotent Law	A. A = A	A + A = A
Inverse Law	AA’ = 0	A + A’ = 1
Commutative Law	AB = BA	A + B = B + A
Associative Law	(AB)C	(A + B) + C = A + (B + C)
Distributive Law	A + BC = (A + B) (A + C)	A (B + C) = AB + AC
Absorption Law	A (A + B) = A	A + AB = A
De Morgan’s Law	(AB)’ = A’ + B’	(A + B)’ = A’B’

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