Element (mathematics)

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In mathematics, an element (or member) of a set is any one of the distinct objects that belong to that set.

Contents

Sets

Writing means that the elements of the set A are the numbers 1, 2, 3 and 4. Sets of elements of A, for example , are subsets of A.

Sets can themselves be elements. For example, consider the set . The elements of B are not 1, 2, 3, and 4. Rather, there are only three elements of B, namely the numbers 1 and 2, and the set .

The elements of a set can be anything. For example, is the set whose elements are the colors red, green and blue.

In logical terms, (xy) ↔ (∀x[Px = y] : x ∈ 𝔇y).

Notation and terminology

The relation "is an element of", also called set 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 includes x" and "A contains x" are also used to mean set membership, although some authors use them to mean instead "x is a subset of A". [2] Logician George Boolos strongly urged that "contains" be used for membership only, and "includes" for the subset relation only. [3]

For the relation ∈ , the converse relationT may be written

meaning "A contains or includes x".

The negation of set membership is denoted by the symbol "∉". Writing

means that "x is not an element of A".

The symbol ∈ was first used by Giuseppe Peano, in his 1889 work Arithmetices principia, nova methodo exposita . [4] Here he wrote on page X:

Signum ∈ significat est. Ita a ∈ b legitur a est quoddam b; …

which means

The symbol ∈ means is. So a ∈ b is read as a is a certain b; …

The symbol itself is a stylized lowercase Greek letter epsilon ("ϵ"), the first letter of the word ἐστί , which means "is". [4]

Character information
Preview
Unicode nameELEMENT OFNOT AN ELEMENT OFCONTAINS AS MEMBERDOES NOT CONTAIN AS MEMBER
Encodingsdecimalhexdechexdechexdechex
Unicode 8712U+22088713U+22098715U+220B8716U+220C
UTF-8 226 136 136E2 88 88226 136 137E2 88 89226 136 139E2 88 8B226 136 140E2 88 8C
Numeric character reference ∈∈∉∉∋∋∌∌
Named character reference ∈, ∈, ∈, ∈∉, ∉, ∉∋, ∋, ∋, ∋∌, ∌, ∌
LaTeX \in\notin\ni\not\ni or \notni
Wolfram Mathematica \[Element]\[NotElement]\[ReverseElement]\[NotReverseElement]

Cardinality of sets

The number of elements in a particular set is a property known as cardinality; 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 set B and set C are both 3. An infinite set is a set with an infinite number of elements, while a finite 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, ...}.

Examples

Using the sets defined above, namely A = {1, 2, 3, 4}, B = {1, 2, {3, 4}} and C = {red, green, blue}, the following statements are true:

Formal relation

As a relation, set membership must have a domain and a range. Conventionally the domain is called the universe denoted U. The range is the set of subsets of U called the power set of U and denoted P(U). Thus the relation is a subset of U x P(U). The converse relation is a subset of P(U) x U.

See also

Related Research Articles

Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics. Unlike axiomatic set theories, which are defined using formal logic, naive set theory is defined informally, in natural language. It describes the aspects of mathematical sets familiar in discrete mathematics, and suffices for the everyday use of set theory concepts in contemporary mathematics.

<span class="mw-page-title-main">Cardinal number</span> Size of a possibly infinite set

In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. In the case of a finite set, its cardinal number, or cardinality is therefore a natural number. For dealing with the case of infinite sets, the infinite cardinal numbers have been introduced, which are often denoted with the Hebrew letter (aleph) marked with subscript indicating their rank among the infinite cardinals.

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In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th-century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of whether number theory is consistent and complete.

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<span class="mw-page-title-main">Union (set theory)</span> Set of elements in any of some sets

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In mathematical logic, a non-standard model of arithmetic is a model of first-order Peano arithmetic that contains non-standard numbers. The term standard model of arithmetic refers to the standard natural numbers 0, 1, 2, …. The elements of any model of Peano arithmetic are linearly ordered and possess an initial segment isomorphic to the standard natural numbers. A non-standard model is one that has additional elements outside this initial segment. The construction of such models is due to Thoralf Skolem (1934).

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References

  1. Weisstein, Eric W. "Element". mathworld.wolfram.com. Retrieved 2020-08-10.
  2. Eric Schechter (1997). Handbook of Analysis and Its Foundations. Academic Press. ISBN   0-12-622760-8. p. 12
  3. George Boolos (February 4, 1992). 24.243 Classical Set Theory (lecture) (Speech). Massachusetts Institute of Technology.
  4. 1 2 Kennedy, H. C. (July 1973). "What Russell learned from Peano". Notre Dame Journal of Formal Logic. Duke University Press. 14 (3): 367–372. doi: 10.1305/ndjfl/1093891001 . MR   0319684.
  5. "Sets - Elements | Brilliant Math & Science Wiki". brilliant.org. Retrieved 2020-08-10.

Further reading