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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 ...
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-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 ...
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.
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 set N of natural numbers is defined in this system as the smallest set containing 0 and closed under the successor function S defined by S(n) = n ∪ {n}. The structure N, 0, S is a model of the Peano axioms (Goldrei 1996). The existence of the set N is equivalent to the axiom of infinity in ZF set theory.
However, when restricted to the context of subsets of a given fixed set , the notion of the intersection of an empty collection of subsets of is well-defined. In that case, if M {\displaystyle M} is empty, its intersection is ⋂ M = ⋂ ∅ = { x ∈ X : x ∈ A for all A ∈ ∅ } {\displaystyle \bigcap M=\bigcap \varnothing =\{x\in X:x\in A ...
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.