set of all even integers, expressed in set-builder notation. In mathematics and more specifically in set theory, set-builder notation is a notation for...

{\displaystyle \{0,1,-1,2,-2,3,-3,\ldots \}.} Set-builder notation specifies a set as being the set of all elements that satisfy some logical formula...

where a is an element of A and b is an element of B. In terms of set-builder notation, that is A × B = { ( a , b ) ∣ a ∈ A    and    b ∈ B } . {\displaystyle...

has blonde hair} denotes the set of everything with blonde hair. This notation is called set-builder notation (or "set comprehension", particularly in...

of the collection M {\displaystyle M} is defined as the set (see set-builder notation) ⋂ A ∈ M A = { x :  for all  A ∈ M , x ∈ A } . {\displaystyle \bigcap...

using the XOR operation ⊕ on the predicates describing the two sets in set-builder notation: A Δ B = { x : ( x ∈ A ) ⊕ ( x ∈ B ) } . {\displaystyle A\mathbin...

the set of elements which are in A, in B, or in both A and B. In set-builder notation, A ∪ B = { x : x ∈ A  or  x ∈ B } {\displaystyle A\cup B=\{x:x\in...

2.  Set-builder notation for a singleton set: { x } {\displaystyle \{x\}} denotes the set that has x as a single element. {□, ..., □} Set-builder notation:...

mathematical set-builder notation (set comprehension) as distinct from the use of map and filter functions. Consider the following example in mathematical set-builder...

expressed notationally as 3 ∈ A {\displaystyle 3\in A} . Writing A = { 1 , 2 , 3 , 4 } {\displaystyle A=\{1,2,3,4\}} means that the elements of the set A are...

concept of a relation, but using more notation (including set-builder notation): A function is formed by three sets, the domain X , {\displaystyle X,} the...

a singleton (also known as a unit set or one-point set) is a set with exactly one element. For example, the set { 0 } {\displaystyle \{0\}} is a singleton...

notations are described in International standard ISO 31-11. Thus, in set builder notation, ( a , b ) = ] a , b [ = { x ∈ R ∣ a < x < b } , [ a , b ) = [ a...

this notation is ambiguous, as in some contexts (for example, Minkowski set operations in functional analysis) it can be interpreted as the set of all...

satisfies Φ {\displaystyle \Phi } may be expressed with the shorthand notation Φ ( x ) = y {\displaystyle \Phi (x)=y} . Another approach is taken by the...

set theory Morse–Kelley set theory Tarski–Grothendieck set theory Ackermann set theory Type theory New Foundations Positive set theory Internal set theory...

a set-theoretic paradox published by the British philosopher and mathematician, Bertrand Russell, in 1901. Russell's paradox shows that every set theory...

the notation 2S denoting the power set P(S) are demonstrated in the below. An indicator function or a characteristic function of a subset A of a set S with...

mathematics, an uncountable set, informally, is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related...

between sets, popularized by John Venn (1834–1923) in the 1880s. The diagrams are used to teach elementary set theory, and to illustrate simple set relationships...

steps of collection of sets, followed by a restriction through separation. All results are also expressed using set builder notation. Firstly, given b {\displaystyle...

In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable. The set of natural numbers (whose existence...

the empty set, but this is now considered to be an improper use of notation. The symbol ∅ is available at Unicode point U+2205 ∅ EMPTY SET. It can be...

In set theory, a universal set is a set which contains all objects, including itself. In set theory as usually formulated, it can be proven in multiple...

} . By the axiom of extensionality this set is unique. We usually denote this set using set-builder notation as B = { x ∈ A | φ ( x ) } {\displaystyle...

any sets whatsoever is called a family of sets, set family, or a set system. Additionally, a family of sets may be defined as a function from a set I {\displaystyle...

mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is countable...

Z notation, a formal notation for specifying objects using Zermelo–Fraenkel set theory and first-order predicate logic Ordinal notation Set-builder notation...

is the set of elements that commute with every element of G. It is denoted Z(G), from German Zentrum, meaning center. In set-builder notation, Z(G) =...

{\displaystyle k} -subsets of an n {\displaystyle n} -element set. In set theory, the notation [ A ] k {\displaystyle [A]^{k}} is also common, especially...