In set theory and its applications to logic, mathematics, and computer science, set-builder notation is a mathematical notation for describing a set by enumerating its elements or stating the properties that its members must satisfy.
Contents
- Sets defined by enumeration
- Sets defined by a predicate
- Specifying the domain
- Examples
- More complex expressions on the left side of the notation
- Equivalent predicates yield equal sets
- Set existence axiom
- Z notation
- Parallels in programming languages
- References
Defining sets by properties is also known as set comprehension, set abstraction or as defining a set's intension.
Set-builder notation is sometimes simply referred to as set notation, although this phrase may be better reserved for the broader class of means of denoting sets.
Sets defined by enumeration
A set is an unordered collection of elements. (An element may also be referred to as a member). An element may be any mathematical entity.
A set can be described directly by enumerating all of its elements between curly brackets, as in the following two examples:
This is sometimes called the "roster method" for specifying a set.
When it is desired to denote a set that contains elements from a regular sequence an ellipses notation may be employed, as shown in the next two examples:
There is no order among the elements of a set, but with the ellipses notation we show an ordered sequence before the ellipsis as a convenient notational vehicle for explaining to a reader which elements are in a set. The first few elements of the sequence are shown then the ellipses indicate that the simplest interpretation should be applied for continuing the sequence. Should no terminating value appear to the right of the ellipses then the sequence is considered to be unbounded.
In each preceding example, each set is described by enumerating its elements. Not all sets can be described in this way, or if they can, their enumeration may be too long or too complicated to be useful. Therefore, many sets are defined by a property that characterizes their elements. This characterization may be done informally using general prose, as in the following example.
However, the prose approach may lack accuracy or be ambiguous. Thus, set builder notation is often used with a predicate characterizing the elements of the set being defined, as described in the following section.
Sets defined by a predicate
Set builder notation can be used to describe sets which are defined by a predicate, rather than explicitly enumerated. In this form, set builder notation has three parts: a variable, a colon or vertical bar separator, and a logical predicate. Thus there is a variable on the left of the separator, and a rule on the right of it. These three parts are contained in curly brackets:
or
The vertical bar, which can also be written as a colon, is a separator which can be read as "such that", "for which", or "with the property that". Φ(x) is said to be the "rule". All values of x for which the predicate holds (is true) belong to the set being defined. All values of x for which the predicate does not hold do not belong to the set. Thus
Specifying the domain
This kind of set builder notation does not necessarily define a set. Russell's paradox shows that the expression "
or adjoining it to the predicate:
The ∈ symbol here denotes set membership, while the
In cases where the set E is clear from context, it may not be explicitly specified. It is common in the literature for an author to state the domain ahead of time, and then not specify it in the set builder notation. For example, an author may say something such as, "Unless otherwise stated, variables are to be taken to be natural numbers."
Examples
The following examples illustrate particular sets defined by set builder notation via predicates. In each case, the domain is specified on the left side of the vertical bar, while the rule is specified on the right side.
More complex expressions on the left side of the notation
An extension of set-builder notation replaces the single variable x with a term that may include one or more variables, combined with functions acting on them. So instead of
For example:
When inverse functions can be explicitly stated, the expression on the left can be eliminated through simple substitution. Consider the example set
Equivalent predicates yield equal sets
Two sets are equal if and only if they have the same elements. Sets defined by set builder notation are equal if and only if their set builder rules, including the domain specifiers, are equivalent. That is
if and only if
Therefore, in order to prove the equality of two sets defined by set builder notation, it suffices to prove the equivalence of their predicates, including the domain qualifiers.
For example,
because the two rule predicates are logically equivalent:
This equivalence holds because, for any real number x, we have
Set existence axiom
In many formal set theories, such as Zermelo–Fraenkel set theory, set builder notation is not part of the formal syntax of the theory. Instead, there is a set existence axiom scheme which states that, if E is a set and Φ(x) is a formula in the language of set theory, then there is a set Y whose members are exactly the elements of E that satisfy Φ:
The set Y obtained from this axiom is exactly the set described in set builder notation as
Z notation
In Z notation, the set of all x in a universe of discourse A that satisfy the condition P(x) is written
In Z, the set membership of an element x in set A is written as
denotes the set of all values F(x), where x is in A and P(x) holds.
Parallels in programming languages
A similar notation available in a number of programming languages (notably Python and Haskell) is the list comprehension, which combines map and filter operations over one or more lists.
In Python, the set-builder's braces are replaced with square brackets, parentheses, or curly braces, giving list, generator, and set objects, respectively. Python uses an English-based syntax. Haskell replaces the set-builder's braces with square brackets and uses symbols, including the standard set-builder vertical bar.
The same can be achieved in Scala using Sequence Comprehensions, where the "for" keyword returns a list of the yielded variables using the "yield" keyword.
Consider these set-builder notation examples in some programming languages:
The set builder notation and list comprehension notation are both instances of a more general notation known as monad comprehensions, which permits map/filter-like operations over any monad with a zero element.