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In Mathematics and theoretical Computer Science , an enumeration of a set is a procedure for listing all members of the set in some definite sequence, or, equivalently, a means of assigning a unique Natural Number to each element of the set.

Formally an enumeration of a set ''S'' is a subset ''K'' of \mathbb{N} (the natural numbers) and a function ''f : K -> S'' that is a Bijection . That is, for every number ''k'' in ''K'' there is exactly one element ''s'' in ''S'' such that ''f(k) = s''.


EXAMPLES


  • The natural numbers are enumerable by the function f(x) = x. In this case f: \mathbb{N} o \mathbb{N} is simply the Identity Function .

  • \mathbb{Z}, the set of Integers is enumerable by


:f(x) := \begin{cases} -(x+1)/2, & \mbox{if } x \mbox{ is odd} \ x/2, & \mbox{if } x \mbox{ is even}. \end{cases}

f: \mathbb{N} o \mathbb{Z} is a bijection since every natural number corresponds to exactly one integer. The following table demonstrates the first five values of this enumeration:








x f(x)
0 0
1 1
2 -1
3 2
4 -2


  • All finite sets are enumerable. Let ''S'' be a finite set with ''n'' elements and let ''K = {1,2,...,n}''. Select any element ''s'' in ''S'' and assign ''f(n) = s''. Now set ''S' = S - {s}'' (where - denotes Set Difference ). Select any element ''s in ''S and assign ''f(n-1) = s'''. Continue this process until all elements of the set have been assigned a natural number. Then f : \{1,2,...,n\} o S' is an enumeration of ''S''.




PROPERTIES


  • There exists an enumeration for a set iff the set is Countable .


  • If a set is enumerable it can have many different enumerations. Consider that the set ''{a,b,c}'' can be enumerated by ''f(1) = a, f(2) = b, f(3) = c'' and also by ''f(1) = c, f(2) = a, f(3) = b''.


  • An enumeration of a set gives a Total Order of that set. Although the order may have little to do with the underlying set it is useful when some order of the set is necessary.