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In mathematics, the distributive property of binary operations is a generalization of the distributive law, ... From the point of view of algebra, ...
The algebra of sets is the set-theoretic analogue of the algebra of numbers. Just as arithmetic addition and multiplication are associative and commutative, so are set union and intersection; just as the arithmetic relation "less than or equal" is reflexive, antisymmetric and transitive, so is the set relation of "subset".
A non-associative algebra [1] (or distributive algebra) is an algebra over a field where the binary multiplication operation is not assumed to be associative.That is, an algebraic structure A is a non-associative algebra over a field K if it is a vector space over K and is equipped with a K-bilinear binary multiplication operation A × A → A which may or may not be associative.
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]
Distributive lattice: a lattice in which each of meet and join distributes over the other. A power set under union and intersection forms a distributive lattice. Boolean algebra: a complemented distributive lattice. Either of meet or join can be defined in terms of the other and complementation.
Elementary algebra, also known as high school algebra or college algebra, [1] ... Brackets can be "multiplied out", using the distributive property. For example, ...
A non-associative algebra [3] (or distributive algebra) over a field K is a K-vector space A equipped with a K-bilinear map. The usage of "non-associative" here is meant to convey that associativity is not assumed, but it does not mean it is prohibited – that is, it means "not necessarily associative".
The Order of Operations emerged progressively over centuries. The rule that multiplication has precedence over addition was incorporated into the development of algebraic notation in the 1600s, since the distributive property implies this as a natural hierarchy.