Editing Naive set theory (section)

== Specifying sets ==
The simplest way to describe a set is to list its elements between curly braces (known as defining a set ''extensionally''). Thus {{math|{{mset|1, 2}}}} denotes the set whose only elements are {{val|1}} and {{val|2}}.
(See [[axiom of pairing]].)
Note the following points:
*The order of elements is immaterial; for example, {{math|1={{mset|1, 2}} = {{mset|2, 1}}}}.
*Repetition ([[multiplicity (mathematics)|multiplicity]]) of elements is irrelevant; for example, {{math|1={{mset|1, 2, 2}} = {{mset|1, 1, 1, 2}} = {{mset|1, 2}}}}.
(These are consequences of the definition of equality in the previous section.)

This notation can be informally abused by saying something like {{math|{{mset|dogs}}}} to indicate the set of all dogs, but this example would usually be read by mathematicians as "the set containing the single element ''dogs''".

An extreme (but correct) example of this notation is {{math|{{mset}}}}, which denotes the empty set.

The notation {{math|{{mset|''x'' : ''P''(''x'')}}}}, or sometimes {{math|{{mset|''x'' |''P''(''x'')}}}}, is used to denote the set containing all objects for which the condition {{mvar|P}} holds (known as defining a set ''intensionally'').
For example, {{math|{{mset|''x'' | ''x'' ∈ '''R'''}}}} denotes the set of [[real number]]s, {{math|{{mset|''x'' | ''x'' has blonde hair}}}} denotes the set of everything with blonde hair.

This notation is called [[set-builder notation]] (or "'''set comprehension'''", particularly in the context of [[Functional programming]]).
Some variants of set builder notation are:
*{{math|{{mset|''x'' ∈ ''A'' | ''P''(''x'')}}}} denotes the set of all {{mvar|x}} that are already members of {{mvar|A}} such that the condition {{mvar|P}} holds for {{mvar|x}}. For example, if {{math|'''Z'''}} is the set of [[integer]]s, then {{math|{{mset|''x'' ∈ '''Z''' | ''x'' is even}}}} is the set of all [[even and odd numbers|even]] integers. (See [[axiom of specification]].)
*{{math|{{mset|''F''(''x'') | ''x'' ∈ ''A''}}}} denotes the set of all objects obtained by putting members of the set {{mvar|A}} into the formula {{mvar|F}}. For example, {{math|{{mset|2''x'' | ''x'' ∈ '''Z'''}}}} is again the set of all even integers. (See [[axiom of replacement]].)
*{{math|{{mset|''F''(''x'') | ''P''(''x'')}}}} is the most general form of set builder notation. For example, {{math|{{mset|''x''&prime;s owner | ''x'' is a dog}}}} is the set of all dog owners.