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Distributive Property
In abstract algebra and formal logic, the distributive property of binary operations generalizes the distributive law from boolean algebra and elementary algebra. In propositional logic, distribution refers to two valid rules of replacement. The rules allow one to reformulate conjunctions and disjunctions within logical proofs. For example, in arithmetic:2 ⋅ (1 + 3) = (2 ⋅ 1) + (2 ⋅ 3), but 2 / (1 + 3) ≠ (2 / 1) + (2 / 3).In the left-hand side of the first equation, the 2 multiplies the sum of 1 and 3; on the right-hand side, it multiplies the 1 and the 3 individually, with the products added afterwards. Because these give the same final answer (8), it is said that multiplication by 2 distributes over addition of 1 and 3
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Distributivism
Distributism
Distributism
(also known as distributionism[1] or distributivism)[2] is an economic ideology that developed in Europe
Europe
in the late 19th and early 20th century based upon the principles of Catholic social teaching, especially the teach
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Ordinal Number
In set theory, an ordinal number, or ordinal, is one generalization of the concept of a natural number that is used to describe a way to arrange a collection of objects in order, one after another. Any finite collection of objects can be put in order just by the process of counting: labeling the objects with distinct whole numbers. Ordinal numbers are thus the "labels" needed to arrange collections of objects in order. An ordinal number is used to describe the order type of a well ordered set (though this does not work for a well ordered proper class)
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Summation
In mathematics, summation (capital Greek sigma symbol: ∑) is the addition of a sequence of numbers; the result is their sum or total. If numbers are added sequentially from left to right, any intermediate result is a partial sum, prefix sum, or running total of the summation. The numbers to be summed (called addends, or sometimes summands) may be integers, rational numbers, real numbers, or complex numbers. Besides numbers, other types of values can be added as well: vectors, matrices, polynomials and, in general, elements of any additive group (or even monoid). For finite sequences of such elements, summation always produces a well-defined sum. The summation of an infinite sequence of values is called a series. A value of such a series may often be defined by means of a limit (although sometimes the value may be infinite, and often no value results at all)
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Difference (mathematics)
Subtraction
Subtraction
is an arithmetic operation that represents the operation of removing objects from a collection. It is signified by the minus sign (−). For example, in the adjacent picture, there are 5 − 2 apples—meaning 5 apples with 2 taken away, which is a total of 3 apples. Therefore, 5 − 2 = 3. Subtraction
Subtraction
represents removing or decreasing physical and abstract quantities using different kinds of objects including negative numbers, fractions, irrational numbers, vectors, decimals, functions, and matrices. Subtraction
Subtraction
follows several important patterns. It is anticommutative, meaning that changing the order changes the sign of the answer. It is not associative, meaning that when one subtracts more than two numbers, the order in which subtraction is performed matters. Subtraction
Subtraction
of 0 does not change a number
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Minuend
Subtraction
Subtraction
is an arithmetic operation that represents the operation of removing objects from a collection. It is signified by the minus sign (−). For example, in the adjacent picture, there are 5 − 2 apples—meaning 5 apples with 2 taken away, which is a total of 3 apples. Therefore, 5 − 2 = 3. Subtraction
Subtraction
represents removing or decreasing physical and abstract quantities using different kinds of objects including negative numbers, fractions, irrational numbers, vectors, decimals, functions, and matrices. Subtraction
Subtraction
follows several important patterns. It is anticommutative, meaning that changing the order changes the sign of the answer. It is not associative, meaning that when one subtracts more than two numbers, the order in which subtraction is performed matters. Subtraction
Subtraction
of 0 does not change a number
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Abstract Algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras. The term abstract algebra was coined in the early 20th century to distinguish this area of study from the other parts of algebra. Algebraic structures, with their associated homomorphisms, form mathematical categories. Category theory
Category theory
is a formalism that allows a unified way for expressing properties and constructions that are similar for various structures. Universal algebra is a related subject that studies types of algebraic structures as single objects
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Boolean Algebras
In abstract algebra, a Boolean algebra
Boolean algebra
or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra
Boolean algebra
can be seen as a generalization of a power set algebra or a field of sets, or its elements can be viewed as generalized truth values. It is also a special case of a De Morgan algebra and a Kleene algebra (with involution). Every Boolean algebra
Boolean algebra
gives rise to a Boolean ring, and vice versa, with ring multiplication corresponding to conjunction or meet ∧, and ring addition to exclusive disjunction or symmetric difference (not disjunction ∨)
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Switching Algebra
In mathematics and mathematical logic, Boolean algebra
Boolean algebra
is the branch of algebra in which the values of the variables are the truth values true and false, usually denoted 1 and 0 respectively
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Vector Addition
In mathematics, physics, and engineering, a Euclidean vector (sometimes called a geometric[1] or spatial vector,[2] or—as here—simply a vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors according to vector algebra. A Euclidean vector
Euclidean vector
is frequently represented by a line segment with a definite direction, or graphically as an arrow, connecting an initial point A with a terminal point B,[3] and denoted by A B → . displaystyle overrightarrow AB
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Logical Equivalence
In logic, statements p displaystyle p and q displaystyle q are logically equivalent if they have the same logical content. This is a semantic concept; two statements are equivalent if they have the same truth value in every model (Mendelson 1979:56). The logical equivalence of p displaystyle p and q displaystyle q is sometimes expressed as p ≡ q displaystyle pequiv q , E p q displaystyle textsf E pq , or p ⟺ q displaystyle piff q . However, these symbols are also used for material equivalence; the proper interpretation depends on the context
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Union (set Theory)
In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection.[1] It is one of the fundamental operations through which sets can be combined and related to each other. For explanation of the symbols used in this article, refer to the table of mathematical symbols.Contents1 Union of two sets 2 Algebraic properties 3 Finite unions 4 Arbitrary unions4.1 Notations5 See also 6 Notes 7 External linksUnion of two sets[edit] The union of two sets A and B is the set of elements which are in A, in B, or in both A and B. In symbols, A ∪ B = x : x ∈ A  or  x ∈ B displaystyle Acup B= x:xin A text or xin B .[2]For example, if A = 1, 3, 5, 7 and B = 1, 2, 4, 6 then A ∪ B = 1, 2, 3, 4, 5, 6, 7
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Intersection (set Theory)
In mathematics, the intersection A ∩ B of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements.[1] For explanation of the symbols used in this article, refer to the table of mathematical symbols.Contents1 Basic definition1.1 Intersecting and disjoint sets2 Arbitrary intersections 3 Nullary intersection 4 See also 5 References 6 Further reading 7 External linksBasic definition[edit]Intersection of three sets:   A ∩ B ∩ C displaystyle ~Acap Bcap C Intersections of the Greek, English and Russian alphabet, considering only the shapes of the letters and ignoring their pronunciationExample of an intersection with setsThe intersection of two sets A and B, denoted by A ∩ B, is the set of all objects that are members of both the sets A and B
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Totally Ordered Set
In mathematics, a linear order, total order, simple order, or (non-strict) ordering is a binary relation on some set X displaystyle X , which is antisymmetric, transitive, and total (this relation is denoted here by infix ≤ displaystyle leq )
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Integer
An integer (from the Latin
Latin
integer meaning "whole")[note 1] is a number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, ​5 1⁄2, and √2 are not. The set of integers consists of zero (0), the positive natural numbers (1, 2, 3, …), also called whole numbers or counting numbers,[1][2] and their additive inverses (the negative integers, i.e., −1, −2, −3, …). This is often denoted by a boldface Z ("Z") or blackboard bold Z displaystyle mathbb Z ( Unicode
Unicode
U+2124 ℤ) standing for the German word Zahlen ([ˈtsaːlən], "numbers").[3][4] Z is a subset of the set of all rational numbers Q, in turn a subset of the real numbers R. Like the natural numbers, Z is countably infinite. The integers form the smallest group and the smallest ring containing the natural numbers
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Greatest Common Divisor
In mathematics, the greatest common divisor (gcd) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers
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