Boolean algebra

Boolean algebra, a branch of mathematics also known as binary algebra, is based on the notion that a variable can only be true or false. It uses only the binary numbers zero and one and is primarily used in computer programming languages, set theory, and statistics. Boolean algebra deals with logical operations rather than the numerical operations involved in elementary algebra but is essentially a combination of logic and algebra.

The use of only binary variables, most commonly presented with the possible values of 1 (“true”) or 0 (“false”), is Boolean algebra’s distinguishing factor.

Boolean algebra lets practitioners use the same algebraic technique they would use to solve normal numerical equations to establish logical relationships. By solving Boolean equations, logicians can more easily see when one combination of propositions logically leads to another. A logical statement, or proposition, can either be true or false, just as an equation with real numbers can be true or false depending on the variable’s value. In Boolean algebra, however, variables do not represent the values that make a statement true but, instead, the statement’s truth or falsehood.

Background

English mathematician George Boole first introduced Boolean algebra in his 1847 book, The Mathematical Analysis of Logic. His goal was to make the rules of logic precise, and he expanded upon his groundbreaking concept in his follow-up book, An Investigation of the Laws of Thought.

When Boole first came on the scene, the disciplines of logic and mathematics had developed distinctly for more than two thousand years. Boole’s major achievement was demonstrating how to unify them. Boole effectively created the field of mathematical logic and set the stage for the long series of developments that led, for example, to universal computation.

Around the time Boole published his first work, Augustus De Morgan also published a book titled Formal Logic that introduced the idea of logical validity. If a statement is logically valid, the conclusion will be true if the assumptions are true. Thus began the binary valuation and labeling of ideas as true or false. The Mathematical Analysis of Logic expanded on the ideas De Morgan’s work covered and added an element of algebraic reasoning to binary logic.

Boolean algebra is fundamental to software and hardware operations used in everyday life and is the basis of the modern digital computer. Using Boolean algebra, computers can perform simple or complex operations. A working knowledge is especially useful to those working in information technology, but even for those who will never formally use Boolean algebra, learning its fundamentals can improve thinking and problem-solving abilities and allow a person to consider alternative processes and potential ways to better understand and simplify them.

The concepts of Boolean algebra are highly technical and are typically taught at colleges and universities across the nation and the globe because emerging technologies require workers who can think and understand the operations of machines developed by new technology.

Boolean algebra initially was purely mathematics, with fewer applications than a branch of math such as differential equations and calculus, which underpin physics. But about one hundred years after Boole introduced his concept, mathematicians and scientists discovered a powerful set of applications for formal logic—in the new field of computing—and placed this abstract mathematical and logical tool at the heart of the modern global economy.

Overview

Whereas elementary algebra is expressed using four basic mathematical operations—addition, subtraction, multiplication, and division—Boolean algebra deals with only three operations: conjunction (AND), disjunction (OR), and negation (NOT). The AND operation determines the conjunction of two variables and is only true when both variables are true. The OR operation determines the disjunction of two variables and is only true if at least one variable is true. The NOT operation determines the negation of one variable and is only true if the variable is false.

Truth tables, a core part of Boolean algebra, are exhaustive lists of all possible combinations of input signals along with the value of the output. They can be used to see the output of different operations given the possible truth of each variable.

In modern Boolean algebra, the plus symbol is used to mean OR, the dot symbol to mean AND, and a bar above a variable to mean NOT.

Three laws of Boolean algebra, or properties, are the same as in basic algebra. One such law is the commutative law, which holds that the order of the variable does not matter in OR and AND operation. For example, A+B = B+A.

Another such law is the associative law, which holds that the order of logic operations is irrelevant and parentheses can be moved when two OR or two AND operations are next to each other.

The third law is the distributive law, which allows the factoring of an expression and is the most practiced and significant law associated with Boolean algebra. It involves two operators—AND and OR—and describes what happens when an AND is distributed over an OR operation and what happens when an OR is distributed over an AND operation.

There also are three other common laws: complement laws, identity laws, and double negation laws. The complement laws hold that a variable AND its negation will produce false, while a variable OR its negation will produce true. The identity laws hold that a term OR’ed with a “0” or AND’ed with a “1” will always equal that term. The double negation law holds that negating a variable twice produces the truth value of the variable.

The variables can be considered placeholders used to create more complex mathematical statements. One variable would have two possibilities: true or false. Two variables become more complicated as both variables could be true, both could be false, or one could be true and one could be false.

Boolean algebra is a complex field but has several practical and somewhat simpler applications in modern life.

It is regularly used in the financial world for the mathematical modeling of market activities. A binary tree, for example, can help to research the price of stock options by representing the range of possible outcomes in the security. For example, in a binomial options pricing model, which has only two possible outcomes (a move up or a move down), the Boolean variable represents an increase or a decrease in the security price. The binomial options pricing model requires that the path of a security’s price gets broken into a series of discrete time ranges.

Boolean algebra is also useful in electrical engineering. By using the variables to represent values of “on” and “off” (or “0” and “1”), Boolean algebra is used to design and analyze the digital switching circuitry found in personal computers, pocket calculators, compact disc players, cellular telephones, and a host of other electronic products.

In fact, one of the first major applications of Boolean algebra came from the 1937 master’s thesis of Claude Shannon, a preeminent twentieth-century mathematician and engineer. Shannon realized that switches in relay networks, such as in a telephone network, or an early proto-computer, could be described by viewing “on” switches has having a Boolean value of “true” and “off” switches as having a Boolean value of “false” The different patterns in which switches are connected to each other correspond to the Boolean operations of “and,” “or,” and “not.”

Shannon’s innovation vastly simplified the design of switch networks, and the techniques Boole and his successors developed provided a mathematical framework allowing for more efficient network layouts.

Other connections between electrical switches and Boolean algebra also exist. Logic gates are at the core of a computer’s central processing unit (CPU) and are physical manifestations of Boolean operators. Logic gates take in one or more electrical Boolean values. A high-voltage wire might represent “true,” while a low-voltage one might represent “false.” The output of the logic gate, calculated using the electronic properties of semiconductors, is the appropriate voltage from the desired Boolean operation. In essence, the ability to perform Boolean operations on various inputs allows a CPU to make decisions on handling those outputs.

Boolean algebra has also been put to effective use in several other modern-day scenarios, including the analysis of flight accidents. In such an analysis, a fault tree might be set up to break down the causes of an undesirable event, such as an aircraft crash, but graphically represented logical expressions are an alternative approach. An event can be defined as a statement that can be true or false, and Boolean algebra rules allow for the restructuring of a fault tree into an equivalent but simpler table. Algebra rules found in Boolean techniques have a practical importance here because a fault tree is a representation of the links between the faults that caused the accident. By translating a fault tree into equivalent Boolean equations (which define events in terms of other basic events), a system failure can be expressed through its basic components.

Numerous researchers and practitioners also have cited the usefulness of Boolean algebra in multiple realms of medicine, including in diagnosing disease by setting its specific diagnostic criteria; in classifying related gene variants; and even in targeting drug delivery directly to tumors.

In diagnostics, when a disease has only a set of symptoms or signs understood to characterize it in an undefined way, Boolean algebra might make such characterization definitive. A study involving irritable bowel syndrome (IBS), for example, showed that a group of doctors acted as if they broadly agreed with an implicit description of the disease that, if made explicit, could lead to diagnosis by logical implication.

Other Boolean algebraic applications in the medical world could be determining the likelihood for certain outcomes of already-diagnosed diseases. Patients themselves might use a Boolean model to estimate likelihoods related to certain conditions when medical experts are not nearby or readily available.

Boolean algebra also has proven useful in defining acute leukemia and complex myeloid neoplasms and, in the 2020s, has been used to develop a simple COVID-19 diagnostic tool that used basic logic gates to determine whether a person had the virus or a related illness such as a flu, cold, or allergy.

Researchers first collected information about the common symptoms of COVID-19 and similar diseases and used the identified symptoms cough, fever, fatigue, loss of taste, and loss of smell as circuit inputs. These were divided into “often,” “sometimes,” “rarely,” and “never.” Options “often” and “sometimes” were considered positive symptoms and “rarely” and “never” were considered negative. Using the logic circuit created from the Boolean expression, researchers successfully predicted the expected disease based on the symptoms. (This experiment, however, had the limitation that some COVID-19 patients are asymptomatic.)

In the treatments realm, a logical system developed by Cole A. DeForest and colleagues at the University of Washington involved a technique in which drugs stored in polymeric hydrogels were released only when Boolean logic-based chemical gates open in response to environmental cues.

Numerous other examples of Boolean algebra in the scientific and medical world exist. In operations research or management science, binary variables and Boolean functions are often used to formulate problems where a number of “go” or “no-go” decisions are needed. Examples include investment decisions in financial management, location decisions in logistics, and assignment decisions in production planning. Most often the variables have to be fixed at values that satisfy constraints expressible as Boolean conditions. This leads to complicated Boolean equations or integer programming problems.

Boolean algebra also can play a role in game theory. Voting games and similar systems of collective choice are often represented by Boolean functions in which the variables are associated with binary alternatives available to the decision makers, and the value of the function indicates the outcome.

Various branches of artificial intelligence also rely on Boolean functions to express deductive reasoning processes or to model primitive cognitive and memorization activities of the brain via neural networks. Boolean functions also can be used to investigate efficient learning strategies or to devise storage and retrieval mechanisms in databases.

Many other practical scenarios can have Boolean models, including cryptography, coding theory, mathematical biology, image processing, theoretical physics, statistics, and more.

Bibliography

Balan, Casandra Venera. “Boolean Algebra Application in the Analysis of Flight Accidents.” INCAS Bulletin7 (4): pp. 41–49. Dec. 2015.

"Boolean Algebra." Britannica, 6 Nov. 2024, www.britannica.com/topic/Boolean-algebra. Accessed 8 Nov. 2024.

“Boolean Algebra.” Tutorials Point, www.tutorialspoint.com/computer‗logical‗organization/boolean‗algebra.htm. Accessed 26 Mar. 2023.

Borman, Stud. “Boolean Logic Used To Trigger Drug Delivery.” Chemical & Engineering News, 22 Jan. 2018, cen.acs.org/articles/96/i4/Boolean-logic-used-trigger-drug.html. Accessed 27 Mar. 2023.

Card, W. I., and R. W. Lucas. “The Logical Description of a Disease Class as a Boolean Function with Special Reference to the Irritable Bowel Syndrome.” National Institutes of Health National Library of Medicine, Mar. 1984, pubmed.ncbi.nlm.nih.gov/6692662/. Accessed 27 Mar. 2023.

Crama, Yves and Peter L. Hammer. “Boolean Functions: Theory, Algorithms, and Applications.” In Encyclopedia of Mathematics and Its Applications. Cambridge University Press, 2011.

Dockter, Janelle. “The History and Rules of Boolean Algebra.” Study.com, 2 Feb. 2022, study.com/learn/lesson/boolean-algebra-rules-examples.html. Accessed 26 Mar. 2023.

Dragovic, Ivana, Nina Turajlic, et al. “A Boolean Consistent Fuzzy Inference System for Diagnosing Diseases and Its Application for Determining Peritonitis Likelihood.” Hindawi, 17 Nov. 2015, www.hindawi.com/journals/cmmm/2015/147947/. Accessed 27 Mar. 2023.

Kenton, Will. “Boolean Algebra.” Investopedia, 29 May 2022, www.investopedia.com/terms/b/boolean-algebra.asp. Accessed 26 Mar. 2023.

Kiersz, Andy. “This Simple Math Concept Went Nowhere for a Century and Then—BOOM—Computers.” Business Insider, 22 July 2014, www.businessinsider.com/boolean-algebra-computers-2014-7. Accessed 26 Mar. 2023.

Kim, D., Chiang Yu, M.N.S., Vicerra, R.R., Naguib, R.N.G. & Concepcion, R.II. “Mathematical Realization of Diagnosing COVID-19 using Boolean Algebra on an Array of Colds-Related Symptoms.” International Research Journal of Science, Technology, Education, and Management, Apr. 2022, irjstem.com/wp-content/uploads/2022/05/IRJSTEM-Volume2‗No1‗Paper1.pdf. Accessed 27 Mar. 2023.

Zugmaier, Gerhard, Sophie Kerkmann, and Franco Locatelli. “Application of Boolean Algebra for Definition of Myeloid Neoplasms.” European Society of Medicine, 31 Jan. 2023. esmed.org/MRA/mra/article/view/3456. Accessed 27 Mar. 2023.