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Boolean logic is a complete system for logical operations. It was named after George Boole, an English mathematician at University College Cork who first defined an algebraic system of logic in the mid 19th century. Boolean logic has many applications in electronics, computer hardware and software. In 1938, Claude Shannon showed how electric circuits with relays were a model for Boolean logic. This fact soon proved enormously consequential with the emergence of the electronic computer. Using the algebra of sets, this article contains a basic introduction to sets, Boolean operations, Venn diagrams, truth tables, and Boolean applications. The Boolean algebra article discusses the algebraic structure of applied Boolean logic. The binary arithmetic article discusses the use of binary numbers in computer systems. Terms Let X be a set: Example Let's imagine that set A contains all even numbers (multiples of two) in "the universe" and set B contains all multiples of three in "the universe". Then the intersection of the two sets (all elements in sets A AND B) would be all multiples of six in "the universe". The complement of set A (all elements NOT in set A) would be all odd numbers in "the universe". Chaining operations together While at most two sets are joined in any Boolean operation, the new set formed by that operation can then be joined with other sets utilizing additional Boolean operations. Using the previous example, we can define a new set C as the set of all multiples of five in "the universe". Thus "sets A AND B AND C" would be all multiples of 30 in "the universe". If more convenient, we may consider set AB to be the intersection of sets A and B, or the set of all multiples of six in "the universe". Then we can say "sets AB AND C" are the set of all multiples of 30 in "the universe". We could then take it a step further, and call this result set ABC. Use of parentheses While any number of logical ANDs (or any number of logical ORs) may be chained together without ambiguity, the combination of ANDs and ORs and NOTs can lead to ambiguous cases. In such cases, parentheses may be used to clarify the order of operations. As always, the operations within the innermost pair is performed first, followed by the next pair out, etc., until all operations within parentheses have been completed. Then any operations outside the parentheses are performed. Properties Let's define symbols for the two primary binary operations as (logical AND/intersection) and (logical OR/union), and for the single unary operation / ~ (logical NOT/complement). We will also use the values 0 (logical FALSE/the empty set) and 1 (logical TRUE/the universe). The following properties apply to both Boolean algebra and Boolean logic: The first three properties define a lattice; the first five define a Boolean algebra. Truth tables For Boolean logic using only two values, 0 and 1, the INTERSECTION and UNION of those values may be defined using truth tables such as these: Other notation Various sets of basic operators may be used to express Boolean logic. AND, OR and NOT are the most intuitive. Mathematicians and engineers often use + for OR and for AND (since in some ways those operations are analogous to addition and multiplication in other algebraic structures and this notation makes it very easy to get sum of products form for people who are familiar with normal algebra). NOT may also be represented by a line drawn above the expression being negated. Programmers will often use the ampersand: &, to denote a logical AND, and the pipe-symbol: |, to denote a logical OR. These symbols are often the standard syntax used for performing bitwise operations. NOT is usually presented using the exclamation mark: !. Another notation uses "meet" for AND and "join" for OR. However, this can lead to confusion, as the term "join" is also commonly used for any Boolean operation which combines sets together, which includes both AND and OR. Basic mathematics use of Boolean terms x + y = 2 AND x - y = 2 x + y < 2 AND x - y < 2 X < 2 OR X = 2 WIDTH = 3 OR WIDTH = -3 English language use of Boolean terms Care should be taken when converting an English sentence into a formal Boolean statement. Many English words have imprecise meanings which may correspond with multiple formal meanings, for example, the word NOT: That could mean that "nothing that glitters is gold" or "some things which glitter are not gold". AND and OR can also be used interchangeably in English, in certain cases: Sometimes the English words AND and OR have the opposite meaning in Boolean logic: Also note that the word OR in English may correspond with either logical OR or logical XOR, depending on the context: The combination AND/OR is sometimes used in English to specify a logical OR, when just using the word OR alone might have been mistaken as meaning logical XOR: A case where this is an issue is when specifications for a computer program or electronic circuit are supplied as an English paragraph describing their function. For example, the statement "the program should verify that the applicant has checked the male or female box", should be taken as an XOR, and a check added to ensure that one, and only one, box is selected. In other cases, the interpretation of English may be less certain, and the author of the specification may need to be consulted to determine their true intent. Digital electronic circuit design Boolean logic is also used for circuit design in electrical engineering; here 0 and 1 may represent the two different states of one bit in a digital circuit, typically high and low voltage. Circuits are described by expressions containing variables, and two such expressions are equal for all values of the variables if, and only if, the corresponding circuits have the same input-output behavior. Furthermore, every possible input-output behavior can be modeled by a suitable Boolean expression. Basic logic gates such as AND, OR, and NOT gates may be used alone, or in conjunction with NAND, NOR, and XOR gates, to control digital electronics and circuitry. Whether these gates are wired in series or parallel controls the precedence of the operations. Database applications Relational databases use SQL, or other database-specific languages, to perform queries, which may contain Boolean logic. For this application, each record in a table may be considered to be an "element" of a "set". For example, in SQL, these SELECT statements are used to retrieve data from tables in the database: Parentheses may be used to explicitly specify the order in which Boolean operations occur, when multiple operations are present: Multiple sets of nested parentheses may also be used, where needed. Any Boolean operation (or operations) which combines two (or more) tables together is referred to as a join, in relational database terminology. In the field of Electronic Medical Records, some software applications use Boolean logic to query their patient databases, in what has been named Concept Processing technology. Search engine queries For this application, each web page on the Internet may be considered to be an "element" of a "set". The various online search engines may each use a different syntax. The syntax used by Google will be described here. "Search term 1" "Search term 2" "Search term 1" OR "Search term 2" -"Search term 1" Interestingly, Google Scholar appears to perform the XOR operation when the OR keyword is used. See also | |||||||
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