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In mathematics, the distributive property of binary operations is a generalization of the distributive law, which asserts that the equality (+) = + is always true in elementary algebra. For example, in elementary arithmetic , one has 2 ⋅ ( 1 + 3 ) = ( 2 ⋅ 1 ) + ( 2 ⋅ 3 ) . {\displaystyle 2\cdot (1+3)=(2\cdot 1)+(2\cdot 3).}
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]
However, there is a natural monad structure on the functor ST if there is a distributive law of the monad S over the monad T. Formally, a distributive law of the monad S over the monad T is a natural transformation: such that the diagrams commute. This law induces a composite monad ST with
Galileo's law of odd numbers. A ramification of the difference of consecutive squares, Galileo's law of odd numbers states that the distance covered by an object falling without resistance in uniform gravity in successive equal time intervals is linearly proportional to the odd numbers. That is, if a body falling from rest covers a certain ...
It also satisfies the distributive law, meaning that (+) = +. These properties may be summarized by saying that the dot product is a bilinear form . Moreover, this bilinear form is positive definite , which means that a ⋅ a {\displaystyle \mathbf {a} \cdot \mathbf {a} } is never negative, and is zero if and only if a = 0 {\displaystyle ...
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).
A rack may be defined as a set with a binary operation such that for every ,, the self-distributive law holds: a ( b c ) = ( a b ) ( a c ) {\displaystyle a\triangleleft (b\triangleleft c)=(a\triangleleft b)\triangleleft (a\triangleleft c)}
The wedge product was introduced originally as an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogues: The magnitude of a 2-blade is the area of the parallelogram defined by and , and, more generally, the magnitude of a -blade is the (hyper)volume of the parallelotope defined by the ...