Element (mathematics)

In mathematics, an element (or member) of a set is any one of the distinct objects that belong to that set.

For elements in category theory, see Element (category theory).

Sets[edit]

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, $C=\{\mathrm {\color {Red}red} ,\mathrm {\color {green}green} ,\mathrm {\color {blue}blue} \}$ is the set whose elements are the colors red, green and blue.

In logical terms, (x ∈ y) ↔ (∀x[P_x = y] : x ∈ 𝔇y).

2 ∈ A

5 ∉ A

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:

Formal relation[edit]

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 $\in$ is a subset of U x P(U). The converse relation $\ni$ is a subset of P(U) x U.

Identity element

Singleton (mathematics)

(1974) [1960], Naive Set Theory, Undergraduate Texts in Mathematics (Hardcover ed.), NY: Springer-Verlag, 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).

Halmos, Paul R.

(2002), "Set Theory", Stanford Encyclopedia of Philosophy, Metaphysics Research Lab, Stanford University

Jech, Thomas

(1972) [1960], Axiomatic Set Theory, NY: Dover Publications, Inc., 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".