Properties of Boolean Algebra
In this article, we will be going through the Properties or Laws of the Boolean algebra. So first we will start our article by defining what are the properties of Boolean Algebra, and then we will go through what are Boolean Addition and Multiplication. Then we will go through the different properties of Boolean Algebra such as Annulment, Identity law, Idempotent law, etc.
Table of Content
- What are The Properties of Boolean Algebra?
- Properties of Boolean Algebra
- Properties of Switching Algebra
- Annulment law
- Identity law
- Idempotent law
- Complement law
- Commutative law
- Associative law
- Distributive law
What are the Properties of Boolean Algebra?
The Boolean Algebra uses sets of rules for analyzing digital gates and circuits which are known as Laws or properties of Boolean Algebra. These laws or properties help to simplify complex Boolean expressions by reducing the number of logic gates required for a given operation.
Variables in the Boolean Algebra are binary which takes only values of either 0 or 1. While expression can be written in an infinite number of variables, each variable can only assume one of these two values. For example, variables labelled as A, B, C, etc., can be utilized to formulate logical expressions such as A + B = C.
Boolean Addition
It is the fundamental operation in Boolean algebra, it is similar to the OR Operation. In digital circuits, it is used to compute the sum term without necessitating the AND operation. The sum term evaluates to true if one or more literals are true, and false if all literals are false. Some of the examples of it are A+B, A’+B, A’+C’.
Boolean Multiplication
It is also one of the fundamental operations in Boolean algebra and it is similar to the AND operation. In a digital circuit, it is used to determine the product term, without using the OR operations. The product term evaluates to true only if all literals are true; otherwise, false. Some of the examples of the AND operation are A.C, A.D.E.
Properties of Boolean Algebra
Switching algebra is also known as Boolean Algebra. It is used to analyze digital gates and circuits It is logical to perform a mathematical operation on binary numbers i.e., on ‘0’ and ‘1’. Boolean Algebra contains basic operators like AND, OR, and NOT, etc. Operations are represented by ‘.’ for AND, and ‘+’ for OR. Operations can be performed on variables that are represented using capital letters eg ‘A’, ‘B’ etc.
Properties of Switching Algebra
The Primary objective of the logic design is to solve the expression to its simplest form. This simplification process is Important to ensure that the final implementation of a logic circuit is as Simple as possible. By reducing complexity, we can increase efficiency and ease of implementation and making the overall design process more simple.
Annulment law
a variable ANDed with 0 gives 0, while a variable ORed with 1 gives 1, i.e.,
A.0 = 0
A + 1 = 1
Identity law
In this law variable remains unchanged it is ORed with ‘0’ or ANDed with ‘1’, i.e.,
A.1 = A
A + 0 = A
Idempotent law
A variable remains unchanged when it is ORed or ANDed with itself, i.e.,
A + A = A
A.A = A
Complement law
In this Law if a complement is added to a variable it gives one, if a variable is multiplied with its complement it results in ‘0’, i.e.,
A + A’ = 1
A.A’ = 0
Double Negation Law
A variable with two negations, its symbol gets cancelled out and original variable is obtained, i.e.,
((A)’)’=A
Commutative law
A variable order does not matter in this law, i.e.,
A + B = B + A
A.B = B.A
Associative law
The order of operation does not matter if the priority of variables are the same like ‘*’ and ‘/’, i.e.,
A+(B+C) = (A+B)+C
A.(B.C) = (A.B).C
Distributive law
This law governs the opening up of brackets, i.e.,
A.(B+C) = (A.B)+(A.C)
(A+B)(A+C) = A + BC
Absorption law
This law involved absorbing similar variables, i.e.,
A.(A+B) = A A + AB
A.(A+B) = A A+ A’B
A.(A+B) = A+B A(A’ + B)
A.(A+B) = AB
De Morgan law
In Demorgan law, the operation of an AND or OR logic circuit is unchanged if all inputs are inverted, the operator is changed from AND to OR, and the output is inverted, i.e.,
(A.B)’ = A’ + B’ (A+B)’ = A’.B’
Consensus theorem
AB + A’C + BC = AB + A’C
Conclusion
In this article, we have gone through different through what are Properties of Boolean Algebra, then we have gone through Boolean Addition and multiplication in brief and after that we have gone through different laws or properties in detail.
Properties of Boolean Algebra – FAQs
What are the basic operators in Boolean Algebra?
The basic operators in Boolean Algebra include AND, OR, and NOT, denoted by ‘.’, ‘+’, and ‘ respectively.
What is the primary aim of logic design?
The primary aim of logic design is to simplify logical expressions to their simplest form ensuring ease of implementation in digital circuits.
What are the applications of Boolean Algebra in digital logic design?
Boolean Algebra have applications in digital logic design, including the design and analysis of digital circuits, such as logic gates, arithmetic circuits, memory units, and processors.