Element (mathematics)

In mathematics, an element, or member, of a set is any one of the distinct objects that make up that set.

Writing A = {1, 2, 3, 4} means that the elements of the set A are the numbers 1, 2, 3 and 4. Sets of elements of A, for example {1, 2}, are subsets of A.

Sets can themselves be elements. For example, consider the set B = {1, 2, {3, 4}}. 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 {3, 4}.

The elements of a set can be anything. For example, C = { red, green, blue }, 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". 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, however 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.

Another possible notation for the same relation is

meaning "A contains x", though it is used less often.

The negation of set membership is denoted by the symbol "∉". Writing

means that "x is not an element of A".

The symbol ϵ was first used by Giuseppe Peano 1889 in his work Arithmetices principia nova methodo exposita. Here he wrote on page X:

"Signum ϵ significat est. Ita a ϵ b legitur a est quoddam b; ..."

which means

"The symbol ϵ means is. So a ϵ 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".

The Unicode characters for these symbols are U+2208 ('element of'), U+220B ('contains as member') and U+2209 ('not an element of'). The equivalent LaTeX commands are "\in", "\ni" and "\notin". Mathematica has commands "\[Element]" and "\[NotElement]".

...
Wikipedia