Axioms of Boolean Algebra
There is a set of axioms or postulates in Boolean algebra that we accept without any proof. We can build a bunch of useful theorems using these axioms. In the following table, I have listed some useful axioms that can be helpful in our further requirements.
x̄=0 (if x=1)
x̄=1 (if x=0)
What are the Boolean Algebra Laws?
The Boolean Algebra laws are a series of rules or expressions that have been developed to assist minimize the number of logic gates required to complete a given logic operation, resulting in a list of functions or theorems that are usually referred to as the Laws of Boolean Algebra. Further, let us discuss these in detail in the upcoming sections.
Law of Complementation
The term complement means ‘to invert’, every variable attains its inverse form after the implementation of the complement operation. The following statements can be made from the law of complementation.
Statement 1: Whenever we apply the AND operation on a Boolean variable and its complement, it results in logic 0.
Statement 2: Whenever we apply the OR operation on a Boolean variable and its complement, it results in logic 1.
Law of Annulment
This Boolean algebra law states that whenever we apply the AND operation on a Boolean variable and 0, it will result in logic 0. After the AND operation with 0, irrespective of the value of the Boolean variable it’s got annulled and results in logic 0.
Similarly, whenever we apply the OR operation on a Boolean variable and 1, it will result in logic 1. After the OR operation with 1, irrespective of the value of the Boolean variable it’s got annulled and results in logic 1.
Law of Identity
The application of OR operation on a Boolean variable and 0 results in the same Boolean variable. Similarly, the application AND operation on a Boolean variable and 1 result in the Boolean variable. In both cases, the result is identical to the Boolean variable.
Idempotent denotes an element of a set that is unchanged in value when multiplied or otherwise operated on by itself. This Boolean algebra law states that for any Boolean variable if we apply AND operation with itself results in the same variable. Similarly, if we apply OR operation on a Boolean variable with itself results in the same variable.
Commutative means the condition that a group of quantities connected by operators gives the same result whatever the order of the quantities involved. If we apply AND operation on two Boolean variables, then there is no significance for the order of application. Similarly in the OR operation of Boolean variables, the order of application is insignificant.
This Boolean algebra law allows the multiplication or factorization outside the Boolean expression. This law is applicable for AND as well as OR operators in Boolean algebra.
X(Y+Z)=X.Y+Y.Z (OR distributive law)
X+(Y.Z)=(X+Y).(Y+Z) (AND distributive law)
This Boolean algebra law allows the grouping of variables. This law is defined for AND as well as OR operators in Boolean algebra.
X+(Y+Z)= (X+Y)+Z= X+Y+Z (OR associative law)
X(Y.Z)= (X.Y)Z= XYZ (AND associative law)
Double Negation Law
This law of Boolean algebra states the double application of inversion on any Boolean variable results in the same variable. It is denoted by the following formula.
X̿ = X
Law of Absorption
This Boolean algebra law helps in the elimination of similar variables; hence it enables the reduction of complex terms into simplified expressions. This law is applicable for both AND as well as OR operators.
X+(X.Y)= (X.1)+(X.Y)= X(1+Y)=X (OR absorption law)
X(X+Y)= (X+0).(X+Y)= X+(0.Y)= X (AND absorption law)
De Morgan’s Theorem
De Morgan’s theorem is a powerful tool of reduction in Boolean algebra. There are two separate statements for theorem.
Statement 1: The complement of the sum of the variables is equal to the product of their individual compliments.
We can prove the above statement using a two-variable truth table as shown below.
The complement of the product of the variables is equal to the sum of their individual complements.
The above statement can be proved by using the two-variable truth table as shown below.
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