# Counting Function

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A counting fuction is a unary integer-valued multiset measure function that represents the total number of members of a multiset.

**AKA:**Set Cardinality, Set Size.**Context:**- Input: a Multiset.
- Output: a Count Value/Cardinal Value (a non-negative integer that represents the maximum number required for a one-to-one mapping to each set member).
- It can be calculated with the Summation Operation of Adding One (1) for every Member of a Multiset.
- It can be represented as |A|.
- It can be an Arity Function.
- It can range from being an Abstract Counting Function to being a Physical Counting Function.
- …

**Example(s):**- Count({}} ⇒ 0; Empty Set.
- Count({X}} ⇒ 1.
- Count({1,A,28,DL}} ⇒ 4.
- Count({b}), [math]\displaystyle{ c }[/math],
*a*, [math]\displaystyle{ b }[/math],*a*}) ⇒ 5. - Count({(
*b*,2), (*c*,1), (*a*,2)}) ⇒ 5. - Count({1,2,3,4,...}) ⇒ Infinity (for an Infinite Set).
- a String Length Function.
- …

**Counter-Example(s):**- an Absolute Frequency Function, e.g. Freq({
*b*, [math]\displaystyle{ c }[/math],*a*, [math]\displaystyle{ b }[/math],*a*},*a*) ⇒ 3. - a Multiset Membership Function, e.g. [math]\displaystyle{ f }[/math]({
*b*, [math]\displaystyle{ c }[/math],*a*, [math]\displaystyle{ b }[/math],*a*}) ⇒ {*b*, [math]\displaystyle{ c }[/math],*a*} - any Binary Set Relation.
- any Union Operation.
- any Distance Function.

- an Absolute Frequency Function, e.g. Freq({
**See:**Counting Task, Summation Function, Small Set, Large Set.

## References

### 2017

- (Wikipedia, 2017) ⇒ https://en.wikipedia.org/wiki/counting Retrieved:2017-2-13.
**Counting**is the action of finding the number of elements of a finite set of objects. The traditional way of counting consists of continually increasing a (mental or spoken) counter by a unit for every element of the set, in some order, while marking (or displacing) those elements to avoid visiting the same element more than once, until no unmarked elements are left; if the counter was set to one after the first object, the value after visiting the final object gives the desired number of elements. The related term*enumeration*refers to uniquely identifying the elements of a finite (combinatorial) set or infinite set by assigning a number to each element.Counting sometimes involves numbers other than one; for example, when counting money, counting out change, "counting by twos" (2, 4, 6, 8, 10, 12, ...), or "counting by fives" (5, 10, 15, 20, 25, ...).

There is archeological evidence suggesting that humans have been counting for at least 50,000 years.

^{[1]}Counting was primarily used by ancient cultures to keep track of social and economic data such as number of group members, prey animals, property, or debts (i.e., accountancy). The development of counting led to the development of mathematical notation, numeral systems, and writing.

- ↑
*An Introduction to the History of Mathematics*(6th Edition) by Howard Eves (1990) p.9

### 2009

- (WordNet, 2009) ⇒ http://wordnetweb.princeton.edu/perl/webwn?s=count
- S: (n) count (the total number counted) "a blood count"
- S: (n) count, counting, numeration, enumeration, reckoning, tally (the act of counting; reciting numbers in ascending order) "the counting continued for several hours"