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For example, the dual of (A & B ∨ C) would be (¬A ∨ ¬B & ¬C). The dual of a formula φ is notated as φ*. The Duality Principle states that in classical propositional logic, any sentence is equivalent to the negation of its dual. [4] [7] Duality Principle: For all φ, we have that φ = ¬(φ*). [4] [7] Proof: By induction on complexity ...
In the mathematical area of order theory, every partially ordered set P gives rise to a dual (or opposite) partially ordered set which is often denoted by P op or P d.This dual order P op is defined to be the same set, but with the inverse order, i.e. x ≤ y holds in P op if and only if y ≤ x holds in P.
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.
Duality principle or principle of duality may refer to: Duality (projective geometry) Duality (order theory) Duality principle (Boolean algebra) Duality principle for sets; Duality principle (optimization theory) Lagrange duality; Duality principle in functional analysis, used in large sieve method of analytic number theory; Wave–particle duality
A set C (blue) and its dual cone C * (red).. A duality in geometry is provided by the dual cone construction. Given a set of points in the plane (or more generally points in ), the dual cone is defined as the set consisting of those points (,) satisfying + for all points (,) in , as illustrated in the diagram.
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)
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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.