Element (mathematics)
Wikipedia, the free encyclopedia - Cite This SourceIn mathematics, the elements or members of a set (or more generally a class) are all those objects which when collected together make up the set (or class).
Set theory and elements
Writing , means that the elements of the set are the numbers 1, 2, 3 and 4. Groups of elements of , for example , are subsets of .Elements can themselves be sets. For example consider the set . The elements of are not 1, 2, 3, and 4. Rather, there are only three elements of , 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.
Notation
The relation "is an element of", also called set membership, is denoted by , and writingmeans that is an element of . Equivalently one can say or write " is a member of ", " belongs to ", " is in ", or " includes ", or " contains ". The negation of set membership is denoted by .
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. In the above examples the cardinality of the set is 4, while the cardinality of the sets and 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, .Examples
Using the sets defined above as- is a member of
- The cardinality of is finite and equal to 6.
- The cardinality of (the prime numbers) is infinite.
References
- Paul R. Halmos 1960, Naive Set Theory, Springer-Verlag, NY, ISBN 0-387-90092-6. "Naive" means that it is not fully axiomatized, not that it is silly or easy (Halmos's treatment is neither).
- Patrick Suppes 1960, 1972, Axiomatic Set Theory, Dover Publications, Inc. NY, ISBN 0-486-61630-4. Both the notion of set (a collection of members), membership or element-hood, the axiom of extension, the axiom of separation, and the union axiom (Suppes calls it the sum axiom) are needed for a more thorough understanding of "set element".
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