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A set of polygons in an Euler diagram This set equals the one depicted above since both have the very same elements.. In mathematics, a set is a collection of different [1] things; [2] [3] [4] these things are called elements or members of the set and are typically mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other ...
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects.Although objects of any kind can be collected into a set, set theory – as a branch of mathematics – is mostly concerned with those that are relevant to mathematics as a whole.
In mathematics, the algebra of sets, not to be confused with the mathematical structure of an algebra of sets, defines the properties and laws of sets, the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions ...
So the intersection of the empty family should be the universal set (the identity element for the operation of intersection), [4] but in standard set theory, the universal set does not exist. However, when restricted to the context of subsets of a given fixed set X {\displaystyle X} , the notion of the intersection of an empty collection of ...
Within set theory, many collections of sets turn out to be proper classes. Examples include the class of all sets (the universal class), the class of all ordinal numbers, and the class of all cardinal numbers. One way to prove that a class is proper is to place it in bijection with the class of all ordinal numbers.
It describes the aspects of mathematical sets familiar in discrete mathematics (for example Venn diagrams and symbolic reasoning about their Boolean algebra), and suffices for the everyday use of set theory concepts in contemporary mathematics. [4] Sets are of great importance in mathematics; in modern formal treatments, most mathematical ...
Class (set theory) – Collection of sets in mathematics that can be defined based on a property of its members; Combinatorial design – Symmetric arrangement of finite sets; δ-ring – Ring closed under countable intersections; Field of sets – Algebraic concept in measure theory, also referred to as an algebra of sets
To investigate the left distributivity of set subtraction over unions or intersections, consider how the sets involved in (both of) De Morgan's laws are all related: () = = () always holds (the equalities on the left and right are De Morgan's laws) but equality is not guaranteed in general (that is, the containment might be strict).