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The 256-element free Boolean algebra on three generators is deployed in computer displays based on raster graphics, which use bit blit to manipulate whole regions consisting of pixels, relying on Boolean operations to specify how the source region should be combined with the destination, typically with the help of a third region called the mask.
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 Boolean algebra, Poretsky's law of forms shows that the single Boolean equation () = is equivalent to () = if and only if =, where represents exclusive or. The law of forms was discovered by Platon Poretsky.
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.
More generally, the primary algebra is "self-dual", meaning that any primary algebra formula has two sentential or Boolean readings, each the dual of the other. Another consequence of self-duality is the irrelevance of De Morgan's laws ; those laws are built into the syntax of the primary algebra from the outset.
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 dual of the dual of any Boolean function is the original function. The dual of a monotone function is monotone. If one is given a monotone Boolean expression, then replacing all conjunctions by disjunctions produces another monotone Boolean expression for the dual function, following De Morgan's laws. However, this will transform the ...
In mathematics, there is an ample supply of categorical dualities between certain categories of topological spaces and categories of partially ordered sets.Today, these dualities are usually collected under the label Stone duality, since they form a natural generalization of Stone's representation theorem for Boolean algebras.