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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 ...
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.
All concrete Boolean algebras satisfy the laws (by proof rather than fiat), whence every concrete Boolean algebra is a Boolean algebra according to our definitions. This axiomatic definition of a Boolean algebra as a set and certain operations satisfying certain laws or axioms by fiat is entirely analogous to the abstract definitions of group ...
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.
LoF (T14–15) proves the primary algebra analog of the well-known Boolean algebra theorem that every formula has a normal form. Let A be a subformula of some formula B . When paired with C3 , J1a can be viewed as the closure condition for calculations: B is a tautology if and only if A and ( A ) both appear in depth 0 of B .
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.
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 .
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