In 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).
Contents |
Set theory and elements
Writing <math>A= \{1, 2, 3, 4 \}</math>, means that the elements of the set <math>A</math> are the numbers 1, 2, 3 and 4. Groups of elements of <math>A</math>, for example <math>\{1, 2 \}</math>, are subsets of <math>A</math>. Elements can themselves be sets. For example consider the set <math>B= \{1, 2, \{3, 4 \} \}</math>. The elements of <math>B</math> are not 1, 2, 3, and 4. Rather, there are only three elements of <math>B</math>, namely the numbers 1 and 2, and the set <math>\{3, 4 \}</math>. The elements of a set can be anything. For example, <math>C=\{ \mbox{red, green, blue} \}</math>, 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 <math>\in</math>, and writing
- <math>x \in A</math>
means that <math>x</math> is an element of <math>A</math>. Equivalently one can say or write "<math>x</math> is a member of <math>A</math>", "<math>x</math> belongs to <math>A</math>", "<math>x</math> is in <math>A</math>", or "<math>A</math> includes <math>x</math>", or "<math>A</math> contains <math>x</math>". The negation of set membership is denoted by <math>\notin</math>.
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 <math>A</math> is 4, while the cardinality of the sets <math>B</math> and <math>C</math> 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, <math>\mathbb{N} = \{ 1, 2, 3, 4 \ldots \}</math>.
Examples
Using the sets defined above as
- <math>2 \in A.</math>
- <math>\{3, 4 \} \in B.</math>
- <math>\{3, 4 \}</math> is a member of <math>B.</math>
- <math>\mbox{Yellow} \notin C.</math>
- The cardinality of <math>D = \{ 2, 4, 6, 8, 10, 12 \}</math> is finite and equal to 6.
- The cardinality of <math>P = \{ 2, 3, 5, 7, 11, 13 \ldots \}</math> (the prime numbers) is infinite.
Footnotes
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".
