# Boolean algebra

Boolean Algebra is used to analyze and simplify the digital logic circuits. It uses only the binary numbers i. It is also called as Binary Algebra or logical Algebra. Boolean algebra was invented by George Boole in Complement of a variable is represented by an overbar.

Thus, complement of variable B is represented as. Logical ANDing of the two or more variable is represented by writing a dot between them such as A.

Sometime the dot may be omitted like ABC. Any binary operation which satisfies the following expression is referred to as commutative operation. Commutative law states that changing the sequence of the variables does not have any effect on the output of a logic circuit. This law states that the order in which the logic operations are performed is irrelevant as their effect is the same.

This law uses the NOT operation. The inversion law states that double inversion of a variable results in the original variable itself. Boolean Algebra Advertisements. Previous Page. Next Page. Previous Page Print Page. Dashboard Logout.Boolean algebra is a branch of algebra wherein the variables are denoted by Boolean values.

True also represented by a 1 and False also represented by a 0. Boolean algebra differs from the mathematical algebraic system with respect to the operations done on its variables. Since this is a new system, there are some new rules and laws that apply.

You wake one morning with the sunshine falling on your face. You drowsily walk to your coffee maker. What is this you see? What happened to your coffee maker?

Examples of simplifying boolean expressions

Everything in the room — from your TV remote to your motivational posters, everything has just two words on them. What the heck! After the initial panic attack spurred by the changed atmosphere, you realize that the world is now a simpler place. A plain combination of just two values powers every system. And binary is the language of this world. This language is governed by Boolean algebra.

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The OR function is similar to binary additionwhereas the AND function is similar to binary multiplication. In digital electronics, circuits involving Boolean operations are represented in Boolean expressions. Boolean algebra helps in simplification of a given logic expression without altering any functionality of any operations or variables. Boole introduced several relationships between the mathematical quantities that possessed only two values: either True or False, which could also be denoted by a 1 or 0 respectively.

This system was later devised as Boolean Algebra. The results of all mathematical operations performed on these values could also possess only two values: 1 or 0. Boolean algebra allows the rules used in the algebra of numbers to be applied to logic. It simplifies Boolean expressions which are used to represent combinational logic circuits. It also helps in minimizing large expressions to equivalent smaller expressions with lesser terms, thus reducing the complexity of the combinational logic circuit it represents, using lesser logic gates for the circuitry.

Additionally, reducing the size of the circuitry also increases the speed of the circuit.In mathematics and mathematical logicBoolean algebra is the branch of algebra in which the values of the variables are the truth values true and falseusually denoted 1 and 0, respectively.

It is thus a formalism for describing logical operationsin the same way that elementary algebra describes numerical operations. It is also used in set theory and statistics.

### Introduction to Boolean Algebra

A precursor of Boolean algebra was Gottfried Wilhelm Leibniz 's algebra of concepts. Leibniz's algebra of concepts is deductively equivalent to the Boolean algebra of sets. Boole's algebra predated the modern developments in abstract algebra and mathematical logic ; it is however seen as connected to the origins of both fields. In fact, M. Stone proved in that every Boolean algebra is isomorphic to a field of sets.

In the s, while studying switching circuitsClaude Shannon observed that one could also apply the rules of Boole's algebra in this setting, [8] and he introduced switching algebra as a way to analyze and design circuits by algebraic means in terms of logic gates.

Shannon already had at his disposal the abstract mathematical apparatus, thus he cast his switching algebra as the two-element Boolean algebra. In modern circuit engineering settings, there is little need to consider other Boolean algebras, thus "switching algebra" and "Boolean algebra" are often used interchangeably.

Efficient implementation of Boolean functions is a fundamental problem in the design of combinational logic circuits. Modern electronic design automation tools for VLSI circuits often rely on an efficient representation of Boolean functions known as reduced ordered binary decision diagrams BDD for logic synthesis and formal verification.

Logic sentences that can be expressed in classical propositional calculus have an equivalent expression in Boolean algebra. Thus, Boolean logic is sometimes used to denote propositional calculus performed in this way.

Although the development of mathematical logic did not follow Boole's program, the connection between his algebra and logic was later put on firm ground in the setting of algebraic logicwhich also studies the algebraic systems of many other logics.

The closely related model of computation known as a Boolean circuit relates time complexity of an algorithm to circuit complexity. Whereas expressions denote mainly numbers in elementary algebra, in Boolean algebra, they denote the truth values false and true.

These values are represented with the bits or binary digitsnamely 0 and 1. A sequence of bits is a commonly used for such functions.

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Another common example is the subsets of a set E : to a subset F of Eone can define the indicator function that takes the value 1 on Fand 0 outside F. The most general example is the elements of a Boolean algebrawith all of the foregoing being instances thereof.

As with elementary algebra, the purely equational part of the theory may be developed, without considering explicit values for the variables. One might consider that only negation and one of the two other operations are basic, because of the following identities that allow one to define conjunction in terms of negation and the disjunction, and vice versa De Morgan's laws :.

The three Boolean operations described above are referred to as basic, meaning that they can be taken as a basis for other Boolean operations that can be built up from them by composition, the manner in which operations are combined or compounded. Operations composed from the basic operations include the following examples:. These definitions give rise to the following truth tables giving the values of these operations for all four possible inputs.

But if x is false, then the value of y can be ignored; however, the operation must return some boolean value and there are only two choices.Boolean algebra is a type of mathematical operation that, unlike regular algebra, works with binary digits bits : 0 and 1. While 1 represents true, 0 represents false.

Computers can perform simple to extremely complex operations with the use of Boolean algebra. Boolean algebra and Boolean operations are the basis for computer logic. Unlike conventional mathematical operations — addition, subtraction, division and multiplication — the operations in Boolean algebra are different and limited in number.

The NOT operation returns the opposite of the value that is provided to it. For example, 1 is the opposite of 0 and vice versa. So there are just two outcomes of the operation. The AND operation returns 1 in case both the inputs are equal to 1. Else, it returns 0. The OR operation returns 1 only if either of the values given to it is 1. Else, it returns a value of 0. Toggle navigation Menu. Home Dictionary Tags Hardware Electronics. Boolean Algebra Last Updated: December 21, Definition - What does Boolean Algebra mean?

Techopedia explains Boolean Algebra Unlike conventional mathematical operations — addition, subtraction, division and multiplication — the operations in Boolean algebra are different and limited in number.

Boolean algebra is named for George Boole, a mathematician who first described it in Share this:. Related Terms. Related Articles. Understanding Bits, Bytes and Their Multiples. From Ada Lovelace to Deep Learning. The Pioneers of Computer Programming.Boolean numbers represent an entirely different system of mathematics from real numbers, binary is nothing more than an alternative notation for real numbers.

Mathematical rules are based on the defining limits we place on the particular numerical quantities dealt with. What most people assume to be self-evident rules of arithmetic—valid at all times and for all purposes—actually depend on what we define a number to be.

Consequently, we must use a different kind of numerical quantity, or object, for AC circuits complex numbersrather than real numbersand along with this different system of numbers comes a different set of rules telling us how they relate to one another. The Greek philosopher Aristotle founded a system of logic based on only two types of propositions: true and false.

Again, the defining limits of propositional values determine the Laws describing their functions and relations. Boole wrote a treatise on the subject intitled An Investigation of the Laws of Thought, on Which Are Founded the Mathematical Theories of Logic and Probabilitieswhich codified several rules of relationship between mathematical quantities limited to one of two possible values: true or false, 1 or 0.

All arithmetic operations performed with Boolean quantities have but one of two possible outcomes: either 1 or 0.

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It is a world in which all other possibilities are invalid by fiat. As one might guess, this is not the kind of math you want to use when balancing a checkbook or calculating current through a resistor.

Just bear in mind that the system of numbers defining Boolean algebra is severely limited in terms of scope, and that there can only be one of two possible values for any Boolean variable: 1 or 0. It should be clearly understood that Boolean numbers are not the same as binary numbers. Whereas Boolean numbers represent an entirely different system of mathematics from real numbers, binary is nothing more than an alternative notation for real numbers.

The two are often confused because both Boolean math and binary notation use the same two ciphers: 1 and 0. The difference is that Boolean quantities are restricted to a single bit either 1 or 0whereas binary numbers may be composed of many bits adding up in place-weighted form to a value of any finite size. Don't have an AAC account?

## Boolean Algebra

Home Textbook Vol. What is the difference between boolean and binary numbers? Does this mean the inviolable and self-evident rules of arithmetic have been violated? His mathematical system became known as Boolean algebra.

Boolean Algebra vs. Boolean Numbers vs. Binary Numbers It should be clearly understood that Boolean numbers are not the same as binary numbers. Published under the terms and conditions of the Design Science License.Boolean algebrasymbolic system of mathematical logic that represents relationships between entities—either ideas or objects.

The basic rules of this system were formulated in by George Boole of England and were subsequently refined by other mathematicians and applied to set theory. Today, Boolean algebra is of significance to the theory of probability, geometry of sets, and information theory. Furthermore, it constitutes the basis for the design of circuits used in electronic digital computers.

In a Boolean algebra a set of elements is closed under two commutative binary operations that can be described by any of various systems of postulates, all of which can be deduced from the basic postulates that an identity element exists for each operation, that each operation is distributive over the other, and that for every element in the set there is another element that combines with the first under either of the operations to yield the identity element of the other.

The ordinary algebra in which the elements are the real numbers and the commutative binary operations are addition and multiplication does not satisfy all the requirements of a Boolean algebra. The advantage of Boolean algebra is that it is valid when truth-values —i. It lends itself to manipulating propositions that are either true with truth-value 1 or false with truth-value 0.

Two such propositions can be combined to form a compound proposition by use of the logical connectives, or operators, AND or OR. The truth-value of the resulting proposition is dependent on the truth-values of the components and the connective employed. For example, the propositions a and b may be true or false, independently of one another.

Boolean algebra. Article Media. Info Print Cite. Submit Feedback. Thank you for your feedback. Home Science Mathematics. The Editors of Encyclopaedia Britannica Encyclopaedia Britannica's editors oversee subject areas in which they have extensive knowledge, whether from years of experience gained by working on that content or via study for an advanced degree See Article History.

Alternative Titles: complemented distributive lattice, complemented lattice. Get exclusive access to content from our First Edition with your subscription. Subscribe today. Learn More in these related Britannica articles:.Boldness has courage, genius and magic in it.

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