Axiom of countable choiceThe axiom of countable choice or axiom of denumerable choice, denoted ACω, is an axiom of set theory that states that every countable collection of non-emptysets must have a choice function.
Axiom of choiceIn mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, one can identify another set containing one element chosen from each set, even if the collection is infinite.Well-ordering theoremIn mathematics, the well-ordering theorem, also known as Zermelo's theorem, states that every set can be well-ordered. A set X is well-ordered by a strict total order if every non-empty subset of X has a least element under the ordering. The well-ordering theorem together with Zorn's lemma are the most important mathematical statements that are equivalent to the axiom of choice (often called AC, see also Axiom of choice § Equivalents).Countable setA mathematical set is countable if either it is finite or it can be put in one to one correspondence with the set of natural numbers. Equivalently, a set is countable if there exists an injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural number, or that the elements of the set can be counted one at a time, although the counting may never finish due to an infinite number of elements.
Ernst ZermeloErnst Friedrich Ferdinand Zermelo (; German:; 27 July 1871 – 21 May 1953) was a German logician and mathematician, whose work has major implications for the foundations of mathematics. He is known for his role in developing Zermelo–Fraenkel axiomatic set theory and his proof of the well-ordering theorem.
Multivalued functionIn mathematics, a multivalued function, multiple-valued function, many-valued function, or multifunction, is a function that has two or more values in its range for at least one point in its domain. It is a set-valued function with additional properties depending on context; though some authors do not distinguish between set-valued functions and multifunctions.
Power setIn mathematics, the power set (or powerset) of a setS is the set of all subsets of S, including the empty set and S itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set. The powerset of S is variously denoted as P(S), 𝒫(S), P(S), P(S){\displaystyle \mathbb {P} (S)}, or 2.
Union (set theory)In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection. It is one of the fundamental operations through which sets can be combined and related to each other. A nullary union refers to a union of zero (0{\displaystyle 0}) sets and it is by definition equal to the empty set. For explanation of the symbols used in this article, refer to the table of mathematical symbols.
Finite setIn mathematics, a finite set is a collection of finitely many different things; the things are called elements or members of the set and are typically mathematical objects, such as numbers, symbols, points in space, lines, other geometric shapes, variables, or other sets. Informally, a finite set is a set which one could in principle count and finish counting. For example, { 2 , 4 , 6 , 8 , 10 } {\displaystyle \{2,4,6,8,10\}} is a finite set with five elements.Differential inclusionIn mathematics, differential inclusions are a generalization of the concept of ordinary differential equation of the form dxdt(t)∈F(t,x(t)),{\displaystyle {\frac {dx}{dt}}(t)\in F(t,x(t)),}where F is a multivalued map, i.e. F(t, x) is a set rather than a single point in Rd{\displaystyle \mathbb {R} ^{d}}.
Well-orderY indicates that the column's property is always true for the row's term (at the very left), while ✗ indicates that the property is not guaranteedin general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric,is indicated by Y in the "Symmetric" column and ✗ in the "Antisymmetric" column, respectively.Converse (logic)In logic and mathematics, the converse of a categorical or implicational statement is the result of reversing its two constituent statements. For the implication P → Q, the converse is Q → P. For the categorical proposition All S are P, the converse is All P are S. Either way, the truth of the converse is generally independent from that of the original statement.
Function (mathematics)Association of one output to each inputIn mathematics, a function from a setX to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y is called the codomain of the function. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of time.*As an Amazon Associate I earn from qualifying purchases.