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...

ordered pairs (a, b) where a is in A and b is in 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...

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...

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...

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...

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

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

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...

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...

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...

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...

I is some, possibly countably or uncountably infinite, indexing set. In set notation, De Morgan's laws can be remembered using the mnemonic "break the...

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...

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

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...

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

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...

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...

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...

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...

element of y {\displaystyle y} ." This sentence can be abbreviated in set-builder notation as: ∃ I ( ∅ ∈ I ∧ ∀ x ( x ∈ I ⇒ ( x ∪ { x } ) ∈ I ) ) . {\displaystyle...

mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle...

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

individual set A {\displaystyle A} , it is simply denoted | A | {\displaystyle |A|} , with a vertical bar on each side; this is the same notation as absolute...

Set-builder notation Set-theoretic topology Simple theorems in the algebra of sets Subset Θ (set theory) Tree (descriptive set theory) Tree (set theory)...

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

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) =...