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Each logic operator can be used in an assertion about variables and operations, showing a basic rule of inference. Examples: The column-14 operator (OR), shows Addition rule: when p=T (the hypothesis selects the first two lines of the table), we see (at column-14) that p∨q=T.
A law of Boolean algebra is an identity such as x ∨ (y ∨ z) = (x ∨ y) ∨ z between two Boolean terms, where a Boolean term is defined as an expression built up from variables and the constants 0 and 1 using the operations ∧, ∨, and ¬. The concept can be extended to terms involving other Boolean operations such as ⊕, →, and ≡ ...
Boole's expansion theorem, often referred to as the Shannon expansion or decomposition, is the identity: = + ′ ′, where is any Boolean function, is a variable, ′ is the complement of , and and ′ are with the argument set equal to and to respectively.
The term "Boolean algebra" honors George Boole (1815–1864), a self-educated English mathematician. He introduced the algebraic system initially in a small pamphlet, The Mathematical Analysis of Logic, published in 1847 in response to an ongoing public controversy between Augustus De Morgan and William Hamilton, and later as a more substantial book, The Laws of Thought, published in 1854.
Boolean function; Boolean-valued function; Boolean-valued model; Boolean satisfiability problem; Boolean differential calculus; Indicator function (also called the characteristic function, but that term is used in probability theory for a different concept) Espresso heuristic logic minimizer; Logical matrix; Logical value; Stone duality; Stone ...
In Boolean algebra, any Boolean function can be expressed in the canonical disjunctive normal form , [1] minterm canonical form, or Sum of Products (SoP or SOP) as a disjunction (OR) of minterms. The De Morgan dual is the canonical conjunctive normal form ( CCNF ), maxterm canonical form , or Product of Sums ( PoS or POS ) which is a ...
In mathematics, a Boolean function is a function whose arguments and result assume values from a two-element set (usually {true, false}, {0,1} or {-1,1}). [1] [2] Alternative names are switching function, used especially in older computer science literature, [3] [4] and truth function (or logical function), used in logic.
In the example above, the unit clause would be added to the partial model; the simplification of the set of clauses would then proceed as above with the difference that the unit clause is now removed from the set. The resulting set of clauses is equivalent to the original one under the assumption of validity of the literals in the partial model.