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A set may have a finite number of elements or be an infinite set. There is a unique set with no elements, called the empty set; a set with a single element is a singleton. Sets are uniquely characterized by their elements; this means that two sets that have precisely the same elements are equal (they are the same set). [6]
Ernst Zermelo, a contributer to modern Set theory, was the first to explicitly formalize set equality in his Zermelo set theory (now obsolete), by his Axiom der Bestimmtheit. [30] Equality of sets is axiomatized in set theory in two different ways, depending on whether the axioms are based on a first-order language with or without equality.
In mathematics, a set A is a subset of a set B if all elements of A are also elements of B; B is then a superset of A. It is possible for A and B to be equal; if they are unequal, then A is a proper subset of B. The relationship of one set being a subset of another is called inclusion (or sometimes containment).
In constructive mathematics, "not empty" and "inhabited" are not equivalent: every inhabited set is not empty but the converse is not always guaranteed; that is, in constructive mathematics, a set that is not empty (where by definition, "is empty" means that the statement () is true) might not have an inhabitant (which is an such that ).
The equivalence class of a set A under this relation, then, consists of all those sets which have the same cardinality as A. There are two ways to define the "cardinality of a set": The cardinality of a set A is defined as its equivalence class under equinumerosity. A representative set is designated for each
In set theory, the axiom of extensionality states that two sets are equal if and only if they contain the same elements. In mathematics formalized in set theory, it is common to identify relations—and, most importantly, functions —with their extension as stated above, so that it is impossible for two relations or functions with the same ...
Thus a set is completely determined by its elements; the description is immaterial. For example, the set with elements 2, 3, and 5 is equal to the set of all prime numbers less than 6. If the sets A and B are equal, this is denoted symbolically as A = B (as usual).
Define the two measures on the real line as = [,] () = [,] for all Borel sets. Then and are equivalent, since all sets outside of [,] have and measure zero, and a set inside [,] is a -null set or a -null set exactly when it is a null set with respect to Lebesgue measure.