2element

2element is a term used to refer to a set that contains exactly two distinct elements. In mathematics, such a set is typically called a two-element set or a 2-element subset. The term emphasizes the cardinality of the collection rather than any particular structure on the elements.

Formally, a set S is a 2element if there exist distinct elements a and b with S = {a,b}. By the definition of sets, the order of elements does not matter, so {a,b} = {b,a}.

Examples of 2element sets include {1,2}, {apple, orange}, and {x, y}. The elements can be numbers, letters, objects, or other sets, as long as exactly two distinct elements are present.

Key properties and relationships: If the underlying universal set has n elements, the number of distinct 2element subsets is the binomial coefficient C(n,2) = n(n−1)/2. If the universe is infinite, there are infinitely many 2element subsets. In graph theory, the 2element subsets of a vertex set correspond to the edges of an undirected graph, with each edge connecting two distinct vertices.

Related concepts include k-element sets (generalizations to sets with k elements) and ordered pairs, which are distinct from 2element sets because order matters for ordered tuples. 2element sets are fundamental in counting, combinatorics, and the study of simple graphs.