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.
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]
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, ...}.
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 × P(U). The converse relation is a subset of P(U) × U.
Jech, Thomas (2002), "Set Theory", Stanford Encyclopedia of Philosophy, Metaphysics Research Lab, Stanford University
Suppes, Patrick (1972) [1960], Axiomatic Set Theory, NY: Dover Publications, Inc., ISBN0-486-61630-4 - 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".