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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 ...
A derived binary relation between two sets is the subset relation, also called set inclusion. If all the members of set A are also members of set B, then A is a subset of B, denoted A ⊆ B. For example, {1, 2} is a subset of {1, 2, 3}, and so is {2} but {1, 4} is not. As implied by this definition, a set is a subset of itself.
The notation {x : P(x)}, or sometimes {x |P(x)}, is used to denote the set containing all objects for which the condition P holds (known as defining a set intensionally). For example, {x | x ∈ R} denotes the set of real numbers, {x | x has blonde hair} denotes the set of everything with blonde hair. This notation is called set-builder ...
Set is the prototype of a concrete category; other categories are concrete if they are "built on" Set in some well-defined way. Every two-element set serves as a subobject classifier in Set. The power object of a set A is given by its power set, and the exponential object of the sets A and B is given by the set of all functions from A to B. Set ...
For real numbers, the product is unambiguous because () = (); hence the notation is said to be well defined. [1] This property, also known as associativity of multiplication, guarantees the result does not depend on the sequence of multiplications; therefore, a specification of the sequence can be omitted.
Set-builder notation can be used to describe a set that is defined by a predicate, that is, a logical formula that evaluates to true for an element of the set, and false otherwise. [2] In this form, set-builder notation has three parts: a variable, a colon or vertical bar separator, and a predicate. Thus there is a variable on the left of the ...
A concept in set theory and logic that categorizes well-ordered sets by their structure, such that two sets have the same order type if there is a bijective function between them that preserves order. ordinal 1. An ordinal is the order type of a well-ordered set, usually represented by a von Neumann ordinal, a transitive set well ordered by ∈. 2.
This article lists mathematical properties and laws of sets, involving 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, and performing calculations, involving these operations and relations.