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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 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 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 ...
Simplification is the process of replacing a mathematical expression by an equivalent one that is simpler (usually shorter), according to a well-founded ordering. Examples include:
In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another, called the modulus of the operation.. Given two positive numbers a and n, a modulo n (often abbreviated as a mod n) is the remainder of the Euclidean division of a by n, where a is the dividend and n is the divisor.
In a non-distributive lattice, there may be elements that are distributive, but not dual distributive (and vice versa). For example, in the depicted pentagon lattice N 5, the element x is distributive, [2] but not dual distributive, since x ∧ (y ∨ z) = x ∧ 1 = x ≠ z = 0 ∨ z = (x ∧ y) ∨ (x ∧ z).
Equivalently, for a distributive lattice, the implication graph of the 2-satisfiability instance can be partitioned into two connected components, one on the positive variables of the instance and the other on the negative variables; the transitive closure of the positive component is the underlying partial order of the distributive lattice.
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 .