Element (mathematics)

In mathematics, an element (or member) of a set is any one of the distinct objects that belong to that set.
Sets.
Writing formula_1 means that the elements of the set are the numbers 1, 2, 3 and 4. Sets of elements of , for example formula_2, are subsets of .
Sets can themselves be elements. For example, consider the set formula_3. 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 formula_4.
The elements of a set can be anything. For example, formula_5 is the set whose elements are the colors , and .
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"". 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"". Logician George Boolos strongly urged that "contains" be used for membership only, and "includes" for the subset relation only.
For the relation ∈ , the converse relation ∈T may be written
The negation of set membership is denoted by the symbol "∉". Writing
The symbol ∈ was first used by Giuseppe Peano, in his 1889 work . Here he wrote on page X:
which means
The symbol ∈ means "is". So a ∈ b is read as a "is a" b; …
The symbol itself is a stylized lowercase Greek letter epsilon ("ϵ"), the first letter of the word , which means "is".
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 "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: