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Encyclopedia > Element (mathematics)

In mathematics, an element (also called a member) is an object contained in a set (or more generally a class). Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. ...

Writing "A = {1, 2, 3, 4}", means that the elements of the set A are the numbers 1, 2, 3 and 4. Groups of elements of A, for example {1, 2}, are subsets of A. A is a subset of B If X and Y are sets and every element of X is also an element of Y, then we say or write: X is a subset of (or is included in) Y; X âŠ† Y; Y is a superset of (or includes) X; Y âŠ‡ X...

Elements can themselves be sets. For example consider the set B = {1, 2, {3, 4} }. 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 {3, 4}.

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

The relation "is an element of", also called set membership, is denoted by "∈", and writing "x ∈ A", means that x is an element of A. Equivalently one can say or write "x is a member of A", "x belongs to A", "x is in A", or A contains x. The negation of set membership, is denoted by "∉". In mathematics, a relation is a generalization of arithmetic relations, such as = and <, which occur in statements, such as 5 < 6 or 2 + 2 = 4. See relation (mathematics), binary relation and relational algebra. ... Negation, in its most basic sense, changes the truth value of a statement to its opposite. ...

Examples (using the sets defined above):

• 2 ∈ A
• {3, 4} ∈ B
• {3, 4} is a member of B
• 3 ∉ B

The number of elements in a particular set is a property known as cardinality, informally this is the size of a set. In the above examples the cardinality of the set A is 4, while the cardinality of the sets B and C is 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 natural numbers. In linguistics, cardinal numbers is the name given to number words that are used for quantity (one, two, three), as opposed to ordinal numbers, words that are used for order (first, second, third). ... Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is...

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 Element (mathematics) - Wikipedia, the free encyclopedia (293 words) In mathematics, an element (also called a member) is an object contained in a set (or more generally a class). The elements of B are not 1, 2, 3, and 4. The number of elements in a particular set is a property known as cardinality, informally this is the size of a set.
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