Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Element (mathematics)
Summary
In mathematics, an element (or member) of a set is any one of the distinct objects that belong to that set.
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 , and .
In logical terms, (x ∈ y) ↔ (∀x[Px = y] : x ∈ Dy).
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
meaning "A contains or includes x".
The negation of set membership is denoted by the symbol "∉". Writing
means that "x is not an element of A".
The symbol ∈ was first used by Giuseppe Peano, in his 1889 work Arithmetices principia, nova methodo exposita. Here he wrote on page X:
Signum ∈ significat est. Ita a ∈ b legitur a est quoddam b; ...
which means
The symbol ∈ means is. So a ∈ b is read as a is a certain b; ...
The symbol itself is a stylized lowercase Greek letter epsilon ("ε"), the first letter of the word ἐστί, which means "is".
Cardinality
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.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.