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In mathematics, the distributive property of binary operations is a generalization of the distributive law, which asserts that the equality ...
In the second step, the distributive law is used to simplify each of the two terms. Note that this process involves a total of three applications of the distributive property. In contrast to the FOIL method, the method using distributivity can be applied easily to products with more terms such as trinomials and higher.
In mathematics, the associative property [1] is a property of some binary operations that means that rearranging the parentheses in an expression will not change the result. In propositional logic , associativity is a valid rule of replacement for expressions in logical proofs .
R is a monoid under multiplication, meaning that: (a · b) · c = a · (b · c) for all a, b, c in R (that is, ⋅ is associative). There is an element 1 in R such that a · 1 = a and 1 · a = a for all a in R (that is, 1 is the multiplicative identity). [b] Multiplication is distributive with respect to addition, meaning that:
With this definition of multiplication and addition, familiar rules for the arithmetic of rational or real numbers continue to hold for complex numbers. More precisely, the distributive property, the commutative properties (of addition and multiplication) hold.
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".
The Egyptians used the commutative property of multiplication to simplify computing products. [7] [8] Euclid is known to have assumed the commutative property of multiplication in his book Elements. [9] Formal uses of the commutative property arose in the late 18th and early 19th centuries, when mathematicians began to work on a theory of ...
Definitions of the appropriate topologies on spaces of test functions and distributions are given in the article on spaces of test functions and distributions. This article is primarily concerned with the definition of distributions, together with their properties and some important examples.