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Similarly, the product of the elements of the empty set should be considered to be one (see empty product), since one is the identity element for multiplication.

A derangement is a permutation of a set without fixed points.

The mathematical symbols employed below are explained here.

For any set A: The connection between the empty set and zero goes further, however: in the standard set-theoretic definition of natural numbers, sets are used to model the natural numbers.

In the von Neumann construction of the ordinals, 0 is defined as the empty set, and the successor of an ordinal is defined as , such that the Peano axioms of arithmetic are satisfied.

In Zermelo set theory, the existence of the empty set is assured by the axiom of empty set, and its uniqueness follows from the axiom of extensionality.

However, there are very many things that 'have no members', in the set-theoretical sense—namely, all non-sets.

This is because there is logically only one way that a set can contain nothing.In axiomatic mathematics, zero is defined as the cardinality of (that is, the number of elements in) the null set.From this starting point, mathematicians can build the set of natural numbers, and from there, the sets of integers and rational numbers.For example, when considered as a subset of the real numbers, with its usual ordering, represented by the real number line, every real number is both an upper and lower bound for the empty set.That is, the least upper bound (sup or supremum) of the empty set is negative infinity, while the greatest lower bound (inf or infimum) is positive infinity.




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