Absorption Identities: In mathematics, absorption identities refer to a pair of equations that describe the interaction between two binary operations, typically addition and multiplication. The identities state that for any elements a and b:
a + (a * b) = a (left absorption)
(a * b) + a = a (right absorption)
These identities indicate that when one operation is applied to the result of the other operation, it "absorbs" or reduces the result back to the original element.
Absorption Identity: See Absorption Identities.
Absorption Law: See Absorption Identities.
Algebra of Random Variables: The algebra of random variables is a mathematical framework that deals with the manipulation and combination of random variables. Random variables are variables whose values are determined by the outcome of a random process or experiment. The algebra of random variables allows for operations such as addition, subtraction, multiplication, and composition of random variables, enabling the analysis of complex probabilistic systems.
Axiom: In mathematics, an axiom is a statement or proposition that is assumed to be true without proof. Axioms serve as the foundation of a particular mathematical system or theory, and other theorems and results are derived from these assumed truths.
Axiom of Choice: The axiom of choice is a fundamental principle in set theory. It states that given any collection of non-empty sets, it is possible to select exactly one element from each set, even if there is no explicit rule for making the selection. The axiom of choice is often used in mathematical proofs, particularly in the field of analysis.
Axiom of Extensionality: The axiom of extensionality is a foundational principle in set theory. It states that two sets are equal if and only if they have the same elements. In other words, sets are determined solely by their elements, and not by the particular way they are described or constructed.
Axiom of Foundation: The axiom of foundation, also known as the axiom of regularity, is an axiom in set theory that helps prevent the existence of certain paradoxical constructions. It states that every non-empty set A contains an element that is disjoint from A, meaning that there is no element in both A and any of its subsets. This axiom ensures that sets cannot contain themselves or create loops of membership.
Axiom of Infinity: The axiom of infinity is an axiom in set theory that asserts the existence of an infinite set. It states that there exists a set that contains the empty set and, for every set x in the set, it also contains the set formed by adding x to itself.
Axiom of Replacement: The axiom of replacement is an axiom in set theory that allows for the construction of new sets by replacing elements of an existing set. It states that if a set A is well-defined and for every element x in A, there exists a unique set y that satisfies a specific property, then the collection of all such y forms a set.
Axiom of Subsets: The axiom of subsets, also known as the axiom of separation or the comprehension axiom, is an axiom in set theory that allows for the creation of subsets based on specific properties or conditions. It states that for any set A and any property P, there exists a subset of A consisting of all elements that satisfy property P.
Axiom of the Empty Set: The axiom of the empty set is an axiom in set theory that asserts the existence of a set with no elements, called the empty set or the null set. It is denoted by the symbol ∅ or {}. The axiom of the empty set is typically used as a foundation for constructing other sets.
Axiom of the Power Set: The axiom of the power set is an axiom in set theory that states for any set A, there exists a set called the power set
of A, which contains all possible subsets of A. The power set of A is denoted by P(A).
Axiom of the Sum Set: The axiom of the sum set is an axiom in set theory that allows for the formation of a set consisting of all elements that belong to any set in a given collection of sets. It states that for any collection of sets, there exists a set called the sum set that contains all elements from those sets.
Axiom of the Unordered Pair: The axiom of the unordered pair is an axiom in set theory that allows for the creation of a set containing two specific elements. It states that for any two sets a and b, there exists a set containing exactly a and b as its elements.
Axiom Schema: An axiom schema, or axiom scheme, is a template or pattern for constructing multiple axioms within a mathematical system. It specifies a general form of axioms that can be instantiated with different specific conditions or variables to generate multiple individual axioms.
Axiomatic Set Theory: Axiomatic set theory is a branch of mathematics that formalizes the study of sets based on a system of axioms and logical rules. The most commonly used axiomatic set theory is the Zermelo-Fraenkel set theory (ZF), which provides a foundation for most of contemporary mathematics.
Axiomatic System: An axiomatic system, also known as a formal system or a deductive system, is a collection of axioms, logical rules, and inference rules that are used to derive theorems and make logical deductions. It provides a formal framework for rigorous mathematical reasoning.
Axioms of Subsets: See Axiom of Subsets.
Categorical Axiomatic System: A categorical axiomatic system is a formal system that is based on category theory, a branch of mathematics that studies abstract structures and relationships between them. In a categorical axiomatic system, the axioms and rules are formulated in terms of category theory concepts and operations.
Congruence Axioms: Congruence axioms are a set of axioms that define the concept of congruence in various mathematical systems, such as geometry and algebra. These axioms specify the properties and relationships of congruent objects, such as congruent angles, line segments, or shapes.
Continuity Axioms: Continuity axioms are a set of axioms that define the concept of continuity in mathematical analysis. They provide conditions that a function must satisfy in order to be considered continuous. Different sets of continuity axioms can be used depending on the specific context or type of functions being considered.
de Morgan's Laws: de Morgan's laws are a pair of logical equivalences that relate the negation of logical statements involving conjunction (AND) and disjunction (OR). The laws are named after the mathematician Augustus de Morgan and are expressed as follows:
1. The negation of a conjunction is equivalent to the disjunction of the negations:
¬(p ∧ q) ≡ ¬p ∨ ¬q
2. The negation of a disjunction is equivalent to the conjunction of the negations:
¬(p ∨ q) ≡ ¬p ∧ ¬q
Eilenberg-Steenrod Axioms: The Eilenberg-Steenrod axioms are a set of properties and principles that characterize homology and cohomology theories in algebraic topology. These axioms provide a framework for studying the algebraic properties of topological spaces and their invariants.
Equidistance Postulate: The equidistance postulate, also known as the postulate of equidistance, is a geometric postulate that states that if a point is equidistant from two other points, then it is also equidistant from all points on
the line segment connecting the two other points. This postulate forms the basis for the concept of perpendicular bisectors and the construction of circles.
Euclid's Postulates: Euclid's postulates are a set of five fundamental assumptions or principles that form the foundation of Euclidean geometry. These postulates were formulated by the ancient Greek mathematician Euclid and include statements about points, lines, and basic geometric operations such as drawing a line segment, extending a line, and constructing circles.
Excision Axiom: The excision axiom is a principle in algebraic topology that deals with the removal of certain subsets from a given space. It states that if two subsets of a space differ only in a "small" set, then their relative homology groups are isomorphic. The excision axiom allows for the simplification and calculation of homology groups by focusing on specific parts of a space.
Field Axioms: Field axioms, also known as the axioms of a field, are a set of properties that define the structure and operations of a field in abstract algebra. A field is a mathematical structure that consists of a set of elements along with two binary operations, addition and multiplication. The field axioms specify the properties that these operations must satisfy, such as commutativity, associativity, existence of inverses, and distributivity.
Hausdorff Axioms: The Hausdorff axioms, also known as the separation axioms, are a set of axioms that define different levels of separation or distinctness between points and sets in topological spaces. The axioms are named after the mathematician Felix Hausdorff and provide conditions that ensure the existence of open sets that separate points or subsets within a given space.
Hilbert's Axioms: Hilbert's axioms are a set of axioms that provide a foundation for Euclidean geometry. They were formulated by the German mathematician David Hilbert as an attempt to rigorously define and derive the basic principles of geometry. Hilbert's axioms cover concepts such as points, lines, planes, congruence, continuity, and incidence relations.
Homotopy Axiom: The homotopy axiom is a fundamental principle in homotopy theory, a branch of algebraic topology. It states that if two continuous maps between topological spaces can be continuously deformed into each other, then they are considered homotopic. Homotopy theory studies the properties of spaces and maps that are preserved under continuous deformations.
Incidence Axioms: Incidence axioms are a set of axioms that define the relationships between points, lines, and planes in projective geometry. These axioms specify the basic properties of incidence, such as the existence of points and lines, the uniqueness of lines through two distinct points, and the existence of intersecting lines.
Induction Axiom: The induction axiom, or the principle of mathematical induction, is a fundamental axiom used in mathematical proofs and reasoning. It allows for the construction of an infinite set by specifying a base case and a rule for generating new elements from existing ones. The induction axiom states that if a property holds for the base case and for every element generated from previous elements, then it holds for all elements in the set.
Kolmogorov's Axioms: Kolmogorov's axioms, also known as the probability axioms, are a set of three axioms that form the foundation of probability theory. These axioms, formulated by the Russian mathematician Andrey Kolmogorov, specify the basic properties and rules of probability, including the non-negativity of probabilities, the additivity of disjoint events, and the normalization condition.
Long Exact Sequence of a Pair Axiom: The long exact sequence of
a pair axiom is a principle in algebraic topology that relates the homology groups of a space and its subspace. It states that given a pair of spaces, there exists a long exact sequence of homology groups that captures the relationships between the homology groups of the pair and their complements.
Ordering Axioms: Ordering axioms, also known as axioms of order, are a set of properties that define a total order relation on a set. A total order is a binary relation that is reflexive, antisymmetric, transitive, and total, meaning that any two elements can be compared. The ordering axioms specify the properties that the total order relation must satisfy, including the existence of a least element (minimum) and a greatest element (maximum).
Parallel Postulate: The parallel postulate, also known as Euclid's fifth postulate, is one of Euclid's postulates in geometry. It states that if a line intersects two other lines and the sum of the interior angles on one side is less than 180 degrees, then the two lines, when extended, will eventually intersect on that side. The parallel postulate distinguishes Euclidean geometry from non-Euclidean geometries.
Peano Arithmetic: Peano arithmetic, also known as first-order arithmetic or the Peano axioms, is a formal system that provides a foundation for the natural numbers and their arithmetic. It was developed by the Italian mathematician Giuseppe Peano and consists of a set of axioms that define the properties and operations of the natural numbers, including addition, multiplication, and induction.
Peano's Axioms: See Peano Arithmetic.
Playfair's Axiom: Playfair's axiom, also known as the axiom of Euclidean geometry, is an alternative formulation of Euclid's parallel postulate. It states that given a line and a point not on the line, there exists exactly one line through the point that is parallel to the given line. Playfair's axiom is equivalent to the parallel postulate and was named after the Scottish mathematician John Playfair.
Postulate: See Euclid's Postulates.
Presburger Arithmetic: Presburger arithmetic, named after the mathematician MojΕΌesz Presburger, is a restricted form of arithmetic that deals with the properties and operations of the natural numbers, including addition and multiplication, but without incorporating the notion of multiplication. It is a decidable theory, meaning that there exists an algorithm to determine the truth or falsehood of any statement within the arithmetic.
Probability Axioms: See Kolmogorov's Axioms.
Proclus' Axiom: Proclus' axiom, also known as the axiom of continuity, is a geometric axiom that extends Euclid's postulates. It states that given a line and a point not on the line, there exists a sequence of points on the line such that each point of the sequence is distinct and closer to the given point than the previous point. Proclus' axiom ensures the existence of infinitely many points on a line segment.
Zermelo-Fraenkel Axioms: The Zermelo-Fraenkel axioms, often abbreviated as ZF, are a set of axioms that provide the foundation for most of modern set theory. They were formulated by the mathematicians Ernst Zermelo and Abraham Fraenkel and include axioms that define the existence of sets, the membership relation, and operations such as union, intersection, and power set. The ZF axioms also include the axiom of choice, which is independent of the other axioms and introduces additional principles for selecting elements from sets.
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