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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 ...
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 ...
Download as PDF; Printable version; ... It has been called the "fundamental theorem of Boolean algebra". [1] ... Dual form There is a dual form of the Shannon ...
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.
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.
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 .
Dual to ∃ is the unary operator ∀, the universal quantifier, defined as ∀x := (∃x ′) ′. A monadic Boolean algebra has a dual definition and notation that take ∀ as primitive and ∃ as defined, so that ∃x := (∀x ′) ′. (Compare this with the definition of the dual Boolean algebra.)
In theoretical computer science, monotone dualization is a computational problem of constructing the dual of a monotone Boolean function.Equivalent problems can also be formulated as constructing the transversal hypergraph of a given hypergraph, of listing all minimal hitting sets of a family of sets, or of listing all minimal set covers of a family of sets.
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