A finite set is understood to be any set with a limited or countable number of elements. Examples of finite sets are the marbles that are contained in a bag, the set of houses in a neighborhood, or the set P formed by the first twenty (20) natural numbers:
P = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}
The set of stars in the universe is surely immense, but it is not known for sure whether it is finite or infinite. However, the set of planets in the solar system is finite.
Figure 1. The set of polygons is finite and the subset of the regular ones as well. (Wikimedia Commons)
The number of elements in a finite set is called its cardinality and for the set P it is denoted as follows: Card (P) or # P. The empty set has zero cardinality and is considered a finite set.
Properties
Among the properties of finite sets are the following:
1- The union of finite sets gives rise to a new finite set.
2- If two finite sets intersect, a new finite set results.
3- A subset of a finite set is finite and its cardinality is less than or equal to that of the original set.
4- The empty set is a finite set.
Examples
There are many examples of finite sets. Some examples include the following:
The set M of the months of the year, which in extended form can be written like this:
M = {January, February, March, April, May, June, July, August, September, October, November, December}, the cardinality of M is 12.
The set S of the days of the week: S = {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}. The cardinality of S is 7.
The set Ñ of the letters of the Spanish alphabet is a finite set, this set by extension is written like this:
Ñ = {a, b, c, d, e, f, g, h, i, j, k, l, m, n, ñ, o, p, q, r, s, t, u, v, w, x, y, z} and its cardinality is 27.
The set V of the vowels in Spanish is a subset of the set Ñ:
V ⊂ Ñ therefore is a finite set.
The finite set V in extensive form is written like this: V = {a, e, i, o, u} and its cardinality is 5.
Sets can be expressed by comprehension. The set F made up of the letters of the word "finite" is an example:
F = {x / x is a letter of the word "finite"}
Said set expressed in extensive form will be:
F = {f, i, n, t, o} whose cardinality is 5 and therefore is a finite set.
More examples
The colors of the rainbow is another example of a finite set, the set C of these colors is:
C = {red, orange, yellow, green, cyan, blue, violet} and its cardinality is 7.
The set of the phases F of the Moon is another example of a finite set:
F = {New moon, first quarter, full moon, last quarter} this set has cardinality 4.
Figure 2. The planets of the solar system form a finite set. (pixabay)
Another finite set is the one formed by the planets of the solar system:
P = {Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, Pluto} of cardinality 9.
Solved Exercises
Exercise 1
The following set A = {x∊ R / x ^ 3 = 27} is given. Express it in words and write it by extension, indicate its cardinality and say whether or not it is finite.
Solution: The set A is the set of real numbers x such that x cubed of as a result 27.
The equation x ^ 3 = 27 has three solutions: they are x1 = 3, x2 = (-3/2 + 3√3 / 2 i) and x3 = (-3/2 - 3√3 / 2 i). Of the three solutions, only x1 is real, while the other two are complex numbers.
Since the definition of set A says that x belongs to the real numbers, then the solutions to the complex numbers are not part of the set A.
The set A expressed extensively is:
A = {3}, which is a finite set of cardinality 1.
Exercise 2
Write in symbolic form (by comprehension) and in extensive form the set B of real numbers that are greater than 0 (zero) and less than or equal to 0 (zero). Indicate its cardinality and whether or not it is finite.
Solution: B = {x∊ R / 0 <x <= 0}
The set B is empty because a real number x cannot be simultaneously greater and less than zero, just as it cannot be 0 and also less than 0.
B = {} and its cardinality is 0. The empty set is a finite set.
Exercise 3
The set S of the solutions of a certain equation is given. The set S by understanding is written like this:
S = {x∊ R / (x-3) (x ^ 2 - 9x + 20) = 0}
Write said set in extensive form, indicate its cardinality and indicate whether or not it is a finite set.
Solution: First, when analyzing the expression that describes the set S, it is obtained that it is a set of real x values that are solutions of the equation:
(x-3) (x ^ 2 - 9x + 20) = 0 (*)
A solution of this equation is x = 3, which is a real number and therefore belongs to S. But there are more solutions that can be obtained by looking for the solutions of the quadratic equation:
(x ^ 2 - 9x + 20) = 0
The above expression can be factored as follows:
(x - 4) (x - 5) = 0
Which leads us to two more solutions of the original equation (*) which are x = 4 and x = 5. In short, equation (*) has as solutions 3, 4 and 5.
The set S expressed in extensive form looks like this:
S = {3, 4, 5}, which has cardinality 3 and is therefore a finite set.
Exercise 4
There are two sets A = {1, 5, 7, 9, 11} and B = {x ∊ N / x is even ^ x <10}.
Write the set B explicitly and find the union with the set A. Also find the intercept of these two sets and conclude.
Solution: set B is made up of natural numbers such that they are even and are also less than the value 10, therefore in extensive set B it is written as follows:
B = {2, 4, 6, 8}
The union of set A with set B is:
AUB = {1, 2, 4, 5, 6, 7, 8, 9, 11}
and the intercept of set A with set B is written like this:
A ⋂ B = {} = Ø is the empty set.
It should be noted that the union and interception of these two finite sets lead to new sets, which in turn are also finite.
References
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