Boolean algebra, sometimes referred to as the algebra of logic, is a two-valued system of algebra that represents logical relationships and operations. English logician and mathematician George Boole, the first to apply these algebraic techniques to the logic process, contended that any logical statement could be assigned a binary value, such as "true/false" or "yes/no." In 1854 Boole wrote An Investigation of the Laws of Thought, in which he discussed ways of reducing logical relationships to simple statements of equality, inequality, inclusion, and exclusion. Boole then showed ways to express these statements symbolically using a binary (two-valued) code, and stated the algebraic rules that governed these logical relationships. This system of mathematical logic came to be known as Boolean algebra.
It was in 1937 that Claude Shannon, then a graduate student at the Massachusetts Institute of Technology, first discussed the connection between electronic circuits and Boolean algebra. At about that time German engineer Konrad Zuse was using Boolean algebra for his mechanical Z1 calculating machine, and late in that decade American physicist John Atanasoff and graduate student Clifford Berry used it in the design of what is considered the first digital electronic computer, the Atanasoff-Berry computer. In the 1930s and 1940s British mathematician Alan Turing also recognized that binary logic was well-suited to the development of digital computers, and in the mid-1940s Hungarian-American mathematician John von Neumann suggested using the binary system for storing computer programs. Thus Boole's work, first systematically interpreted by Shannon, provided one of the critical pioneering tools for the development of computer science, including modern information processing and the design of computer circuitry.
The binary language used in today's computers reflects Boole's binary logic. To begin with, nearly all modern computers operate solely on the binary numbers "1" and "0." These two digits may be manipulated not only as numbers but as logical values: for example, "1 = TRUE" and "0 = FALSE." All the instructions that direct a computer's operation exist as a sequence of such binary digits or bits (0s and 1s). (Note that when a computer user inputs a decimal number into a computer from a keyboard, say "9," the computer must convert it into a binary number such as "1001" for internal purposes.) These binary digits (or logical variables) are processed in the machine as distinct voltage states in tiny electronic circuits known as or logic gates A logic gate only recognizes two varieties of input, high-voltage (value of 1 or TRUE ) and low-voltage (value of 0 or FALSE). Each logic gate takes in two or more bits in the form of such voltages, combines them according to a built-in rule, and produces a single high-voltage or low-voltage logical conclusion (output). The basic logic gates are AND, OR, and NOT; these gates, used in differing combinations, allow the computer to execute all its operations. A basic AND gate takes the value of two input bits and tests them to see if they are both equal to 1. If they are, the output from the AND gate is a 1. If they are not, the AND gate will output a 0. An OR gate tests two input bits to see if or either of the bits is equal to 1. If so, the gate outputs a 1; if not, it outputs a 0. A NOT gate simply negates the input bit, so an input of 1 results in an output of 0, and vice versa.
As already implied, the binary 0 and 1 states used by computers are naturally related to the "true" and "false" of formal logic. For instance, the logical operation built into an AND gate is equivalent to the logic symbol "." Binary logical variables (e.g., p and q) can be employed either in Boolean statements or built into logic gates: thus "p AND q" in logic circuitry is equivalent to "pq" in logic notation. The same is true for the OR gate and "∨"; the statement "p OR q" is equivalent to "p ∨ q."
Combinations of logic gates can be used to execute operations on data. A body of logic gates wired together form a logic circuit. The output or outputs of a logic circuit can either provide input to another circuit or circuits or produce the final result of an operation. Extremely complex operations can be performed using combinations of the AND, OR, and NOT functions. In fact, or all the operations carried out by a digital computer are achieved by suitably combining these three elementary functions.
Boolean algebra has many practical applications in the sciences, especially in circuit theory and (as we have seen) computer science. As an example of a Boolean algebra application in circuit theory, let p and q denote two propositions (declarative sentences that are either true or false but not both true and false at the same time). If each of the propositions p and q is associated with a switch that will be closed if the proposition is "true" and open if the proposition is "false," then the combined proposition pq (p AND q) may be instantiated by connecting the switches in series--that is, so that the output of one switch is the input of the other. Current can flow through the combined circuit if and only if both switches are closed, that is, if both p and q are "true." Similarly, a circuit with two switches connected in parallel (side by side, so that both contribute to the output simultaneously) can be used to represent the statement p ∨ q (p OR q). In this case current will flow if either switch is closed or if both switches are closed, that is, if p or q or both are "true." More complicated statements are associated with more complex circuits. In computer hardware, "switches" are realized as transistors that turn extremely tiny currents on and off.
In the field of computer science binary logic is called digital logic (a set of rules for representing the relationships and interactions among numbers, symbols, words, and other data stored or entered in a computer's memory). Digital logic is at the heart of the operation of all modern digital computers.
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