Family of sets in the context of Diagram (category theory)

In 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, it is possible to construct a new set by choosing one element from each set, even if the collection is infinite. Formally, it states that for every indexed family ${\displaystyle (S_{i})_{i\in I}}$ of nonempty sets, there exists an indexed set ${\displaystyle (x_{i})_{i\in I}}$ such that ${\displaystyle x_{i}\in S_{i}}$ for every ${\displaystyle i\in I}$. The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem.The axiom of choice is equivalent to the statement that every partition has a transversal.

In many cases, a set created by choosing elements can be made without invoking the axiom of choice, particularly if the number of sets from which to choose the elements is finite, or if a canonical rule on how to choose the elements is available — some distinguishing property that happens to hold for exactly one element in each set. An illustrative example is sets picked from the natural numbers. From such sets, one may always select the smallest number, e.g. given the sets {{4, 5, 6}, {10, 12}, {1, 400, 617, 8000}}, the set containing each smallest element is {4, 10, 1}. In this case, "select the smallest number" is a choice function. Even if infinitely many sets are collected from the natural numbers, it will always be possible to choose the smallest element from each set to produce a set. That is, the choice function provides the set of chosen elements. But no definite choice function is known for the collection of all non-empty subsets of the real numbers. In that case, the axiom of choice must be invoked.

In mathematics, particularly in combinatorics, given a family of sets, here called a collection C, a transversal (also called a cross-section) is a set containing exactly one element from each member of the collection. When the sets of the collection are mutually disjoint, each element of the transversal corresponds to exactly one member of C (the set it is a member of). If the original sets are not disjoint, there are two possibilities for the definition of a transversal:

• One variation is that there is a bijection f from the transversal to C such that x is an element of f(x) for each x in the transversal. In this case, the transversal is also called a system of distinct representatives (SDR).
• The other, less commonly used, does not require a one-to-one relation between the elements of the transversal and the sets of C. In this situation, the members of the system of representatives are not necessarily distinct.

In computer science, computing transversals is useful in several application domains, with the input family of sets often being described as a hypergraph.

In mathematics, the power set (or powerset) of a set S 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), ${\displaystyle \mathbb {P} (S)}$, or 2.Any subset of P(S) is called a family of sets over S.

In mathematics, a field of sets is a mathematical structure consisting of a pair ${\displaystyle (X,{\mathcal {F}})}$ consisting of a set ${\displaystyle X}$ and a family ${\displaystyle {\mathcal {F}}}$ of subsets of ${\displaystyle X}$ called an algebra over ${\displaystyle X}$ that contains the empty set as an element, and is closed under the operations of taking complements in ${\displaystyle X,}$ finite unions, and finite intersections.

Fields of sets should not be confused with fields in ring theory nor with fields in physics. Similarly the term "algebra over ${\displaystyle X}$" is used in the sense of a Boolean algebra and should not be confused with algebras over fields or rings in ring theory.