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The FOIL method is a special case of a more general method for multiplying algebraic expressions using the distributive law. The word FOIL was originally intended solely as a mnemonic for high-school students learning algebra. The term appears in William Betz's 1929 text Algebra for Today, where he states: [2]
In approximate arithmetic, such as floating-point arithmetic, the distributive property of multiplication (and division) over addition may fail because of the limitations of arithmetic precision. For example, the identity 1 / 3 + 1 / 3 + 1 / 3 = ( 1 + 1 + 1 ) / 3 {\displaystyle 1/3+1/3+1/3=(1+1+1)/3} fails in decimal arithmetic , regardless of ...
An element x is called a dual distributive element if ∀y,z: x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z). In a distributive lattice, every element is of course both distributive and dual distributive. In a non-distributive lattice, there may be elements that are distributive, but not dual distributive (and vice versa).
An alternative way of stating the same fact is that every distributive lattice is a subdirect product of copies of the two-element chain, or that the only subdirectly irreducible member of the class of distributive lattices is the two-element chain. As a corollary, every Boolean lattice has this property as well. [6]
This is about lattice theory.For other similarly named results, see Birkhoff's theorem (disambiguation).. In mathematics, Birkhoff's representation theorem for distributive lattices states that the elements of any finite distributive lattice can be represented as finite sets, in such a way that the lattice operations correspond to unions and intersections of sets.
A modular lattice of order dimension 2. As with all finite 2-dimensional lattices, its Hasse diagram is an st-planar graph. In the branch of mathematics called order theory, a modular lattice is a lattice that satisfies the following self-dual condition, Modular law a ≤ b implies a ∨ (x ∧ b) = (a ∨ x) ∧ b
A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra.It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet).
The principle of distributivity states that the algebraic distributive law is valid, where both logical conjunction and logical disjunction are distributive over each other so that for any propositions A, B and C the equivalences () and