Inmathematics, anelement (ormember)of aset is any one of thedistinctobjects that belong to that set. For example, given a set calledA containing the first four positiveintegers(), one could say that "3 is an element ofA", expressed notationally as.
Writing means that the elements of the setA are the numbers 1, 2, 3 and 4. Sets of elements ofA, for example, aresubsets ofA.
Sets can themselves be elements. For example, consider the set. The elements ofB arenot 1, 2, 3, and 4. Rather, there are only three elements ofB, namely the numbers 1 and 2, and the set.
The elements of a set can be anything. For example the elements of the set are the color red, the number 12, and the setB.
In logic, a set can be defined in terms of the membership of its elements as. This basically means that there is a generalpredication of x called membership that is equivalent to the statement ‘x is a member of y if and only if, for all objects x, the general predication of x is identical to y, where x is a member of thedomain of y.’ The expression x ∈ 𝔇y makes this definition well-defined by ensuring that x is a bound variable in its predication of membership in y.
In this case, the domain of Px, which is the set containing all dependent logical values x that satisfy the stated conditions for membership in y, is called the Universe (U) of y. The range of Px, which is the set of all possible dependent set variables y resulting from satisfaction of the conditions of membership for x, is thepower set of U such that thebinary relation of the membership of x in y is any subset of thecartesian product U × 𝒫(U) (the Cartesian Product of set U with the Power Set of U).
Thebinary relation "is an element of", also calledset membership, is denoted by the symbol "∈". Writing
means that "x is an element of A".[1] Equivalent expressions are "x is a member of A", "x belongs to A", "x is in A" and "x lies in A". The expressions "A includesx" and "A containsx" are also used to mean set membership, although some authors use them to mean instead "x is asubset of A".[2] LogicianGeorge Boolos strongly urged that "contains" be used for membership only, and "includes" for the subset relation only.[3]
For the relation ∈ , theconverse relation ∈T may be written
meaning "A contains or includesx".
Thenegation of set membership is denoted by the symbol "∉". Writing
means that "x is not an element of A".
The symbol ∈ was first used byGiuseppe Peano, in his 1889 workArithmetices principia, nova methodo exposita.[4] Here he wrote on page X:
Signum∈ significat est. Itaa∈ b legitur a est quoddam b; …
which means
The symbol ∈ meansis. Soa ∈b is read as ais a certain b; …
The symbol itself is a stylized lowercase Greek letterepsilon ("ϵ"), the first letter of the wordἐστί, which means "is".[4]
| Preview | ∈ | ∉ | ∋ | ∌ | ||||
|---|---|---|---|---|---|---|---|---|
| Unicode name | ELEMENT OF | NOT AN ELEMENT OF | CONTAINS AS MEMBER | DOES NOT CONTAIN AS MEMBER | ||||
| Encodings | decimal | hex | dec | hex | dec | hex | dec | hex |
| Unicode | 8712 | U+2208 | 8713 | U+2209 | 8715 | U+220B | 8716 | U+220C |
| UTF-8 | 226 136 136 | E2 88 88 | 226 136 137 | E2 88 89 | 226 136 139 | E2 88 8B | 226 136 140 | E2 88 8C |
| Numeric character reference | ∈ | ∈ | ∉ | ∉ | ∋ | ∋ | ∌ | ∌ |
| Named character reference | ∈, ∈, ∈, ∈ | ∉, ∉, ∉ | ∋, ∋, ∋, ∋ | ∌, ∌, ∌ | ||||
| LaTeX | \in | \notin | \ni | \not\ni or \notni | ||||
| Wolfram Mathematica | \[Element] | \[NotElement] | \[ReverseElement] | \[NotReverseElement] | ||||
Using the sets defined above, namelyA = {1, 2, 3, 4},B = {1, 2, {3, 4}} andC = {red, 12,B}, the following statements are true:
The number of elements in a particular set is a property known ascardinality; informally, this is the size of a set.[5] In the above examples, the cardinality of the set A is 4, while the cardinality of setB and setC are both 3. An infinite set is a set with an infinite number of elements, while afinite set is a set with a finite number of elements. The above examples are examples of finite sets. An example of an infinite set is the set of positive integers{1, 2, 3, 4, ...}.
As arelation, set membership must have a domain and a range. Conventionally the domain is called theuniverse denotedU. The range is the set ofsubsets ofU called thepower set ofU and denoted P(U). Thus the relation is a subset ofU × P(U). The converse relation is a subset ofP(U) ×U.
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