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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]
The map φ defined by φ(y) = x ∨ y is a lattice homomorphism from L to the upper closure ↑x = { y ∈ L: x ≤ y}; The binary relation Θ x on L defined by y Θ x z if x ∨ y = x ∨ z is a congruence relation, that is, an equivalence relation compatible with ∧ and ∨. [3] In an arbitrary lattice, if x 1 and x 2 are distributive ...
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 ...
The set of all functions f: X → X is called the full transformation semigroup [6] or symmetric semigroup [7] on X. (One can actually define two semigroups depending how one defines the semigroup operation as the left or right composition of functions. [8]) Composition of a shear mapping (red) and a clockwise rotation by 45° (green). On the ...
Distributivity, a property of binary operations that generalises the distributive law from elementary algebra; Distribution (number theory) Distribution problems, a common type of problems in combinatorics where the goal is to enumerate the number of possible distributions of m objects to n recipients, subject to various conditions; see ...
In Cartesian coordinates, the divergence of a continuously differentiable vector field = + + is the scalar-valued function: = = (, , ) (, , ) = + +.. As the name implies, the divergence is a (local) measure of the degree to which vectors in the field diverge.
Xavier Lucas had 12 tackles and one interception for the Wisconsin Badgers in the 2024 college football season. (Jordon Kelly/Getty Images) (Icon Sportswire via Getty Images)
In category theory, an abstract branch of mathematics, distributive laws between monads are a way to express abstractly that two algebraic structures distribute one over the other. Suppose that ( S , μ S , η S ) {\displaystyle (S,\mu ^{S},\eta ^{S})} and ( T , μ T , η T ) {\displaystyle (T,\mu ^{T},\eta ^{T})} are two monads on a category C .