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Cardinality in set theory

Let’s look at the concept of a sequence or a series. These are functions ‘from a subset of the natural numbers to a set of things”(Steinhart, p. 60). The domain of a sequence is typically a subset of the number line representing all natural numbers.

“If S is a sequence from {0,…n} to the set of things T, then S(n) is the n-th item in the sequence (it is the n-th item in T as ordered by S). We use a special notation and write S(n) as Sn[in Steinhart’s text, this is a subscript]. We write the sequence as {S0,…Sn} or sometimes as {Sn} [as before, in Steinhart’s text, 0 and n are subscripts].

Suppose we have a sequence. We add up its numbers in sequential order. “We use a variable I to range over the sequence. The notation “for I varying from 0 to n” means that I takes on all the values from 0 to n in order. To say that I varies from 0 to 3 means that I takes on the values 0, 1, 2, 3,. The sum, for I varying from 0 to 3 of Si[in Steinhart’s text, this is a subscript] is S0 + S1 + S2 + S3”(Steinhart, p. 60).

Note that the numbers paired with S, in Steinhart’s text, are all subscripts. For the variable I to range over the sum of the members of S simply means that we add all of them together.

If we wanted to formally represent such a sequential sum, we would write “the sum, for I varying from 0 to n, of Si[in Steinhart’s text, this is a subscript] =”(Steinhart, p. 60)

Following the equals sign, Steinhart provides us with a large sigma with an n on top of it, an Si [with a subscript] to the right of it, and I = 0 under it. The reader would do well to write this down, as it is difficult to represent in the format I have available here.

We can also represent the union of such a sequence.

“the union, for I varying from 0 to n, of Si =”(Steinhart, p. 61).

Following the equals sign, we have a large U for Union, with an n on top of it, an Si to the right of it, and I = 0 under it(Steinhart, p. 61). The same is true of intersections, except we have a large upside down U to represent intersections, instead of unions.

The cardinality of a set is simply the number of a set’s members. It can be represented |S| or #S, though the former is more common. “The cardinality of a set is n iff there exists a 1-1 correspondence (a bijection) between S and the set of numbers less than n”(Steinhart, p. 61).

Let’s look at some examples. Suppose we have a set of numbers less than 1. This set is {0}. Let’s call this set A. There is a 1-1 correspondence between set A and its element, {0}(Steinhart, p. 61). The cardinality of the set would be 1, because there is only one member. Let’s look at another set, B. This set is the set of numbers less than 2. The elements of this set are {1, 0}. Therefore, the set exhibits a 1-1 correspondence with its elements {1, 0}. Its cardinality is a value of 2 because it has two elements. We can represent the correspondence between the set of sets {A, B} and the cardinality of both {0, 1} by writing A  0, B  1 (Steinhart, p. 61). Therefore, the cardinality of {A, B} is 2(Steinhart, p. 61). Let’s see what happens when we add another set to the mix.

“The set of numbers less than 3 is {0, 1, 2}. There is a correspondence between {A, B, C} and {0, 1, 2}. This is A  0 and B --> 1 and C --> 2. So the cardinality of {A, B, C} is 3”(Steinhart, p. 61). We say that two sets are equicardinal if they possess the same cardinality. In Steinhart’s example, the set of fingers on my right hand are equicardinal to the set of fingers on my left hand (Steinhart, p. 61).

We are familiar with cardinality in everyday life. We use the concept to find averages. Indeed, the very definition of an average is the sum of a set of numbers divided by the cardinality of the set. We can represent this formally, as Steinhart points out, as follows:

Average(S) = [Sigma]S / |S|

Steinhart, Eric. "More Precisely: The Mathematics You Need To Do Philosophy." Broadview Press, 2009.