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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.
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 following proposition says that for any set , the power set of , ordered by inclusion, is a bounded lattice, and hence together with the distributive and complement laws above, show that it is a Boolean 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".
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).
Every Heyting algebra is a distributive lattice. Especially this includes all locales and hence all open set lattices of topological spaces. Also note that Heyting algebras can be viewed as Lindenbaum algebras of intuitionistic logic, which makes them a special case of the first example. Every totally ordered set is a distributive lattice with ...
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