Finite Sets

A finite set is a collection of elements that can be counted one by one until reaching an end. This means that the process of listing its members terminates at some point, unlike unbounded collections such as the integers or real numbers.

In other words, a set A is finite when its cardinality (the total number of elements it contains) is equal to a non-negative integer n. It is commonly denoted as |A| = n, where n ≥ 0.

To formalize this concept, we say that a non-empty set A is finite if it is possible to establish a one-to-one correspondence (or bijection) between its elements and the reference set {1, 2, 3, ..., n} for some positive integer n. Establishing this correspondence means that each element of A is assigned a unique, ordered position from 1 to n, leaving no elements out and repeating none.

Conversely, when a set fails to meet this condition, it is classified as an infinite set. In such sets, it is impossible to finish counting the elements or to match them precisely with a finite list {1, 2, 3, ..., n}, as happens with the set of natural numbers or the points on a line.

Examples

Below, we analyze different finite collections, ranging from everyday groupings written in roster form to numerical sets expressed in set-builder notation:

1. If we consider the days of the week, we define the set D = {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}. Counting its elements confirms that the process terminates exactly, yielding a cardinality of |D| = 7.
2. Classifying the vowels in the English alphabet yields the set V = {a, e, i, o, u}. Since it contains exactly five distinct letters, it is a finite collection with |V| = 5.
3. On a calendar, we group the months of the year into the set M = {January, February, March, ..., December}. The list of its members has a fixed count, establishing that |M| = 12.
4. Consider the digits of the base-10 system written in set-builder notation as S = {x ∈ Z | 0 ≤ x ≤ 9}. Writing its values in roster form yields S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, showing that it is a finite set with cardinality |S| = 10.
5. Within the domain of integers, consider the set P = {x ∈ Z | x is prime ∧ 2 ≤ x < 20}. Listing the prime numbers that satisfy this double inequality gives P = {2, 3, 5, 7, 11, 13, 17, 19}, confirming that its cardinality is |P| = 8.
6. Finding the real values that satisfy a quadratic equation yields the solution set C = {x ∈ R | x² - 9 = 0}. Solving the equation reveals only two possible values, so C = {-3, 3} and its cardinality is |C| = 2.

Properties

Below are the most important properties of finite sets.

1) Finiteness of the empty set: The empty set Ø is finite by definition, as it contains no elements and its cardinality is zero.

|Ø| = 0

2) Subsets of a finite set: Any subset taken from a finite set is necessarily finite, and its cardinality can never exceed that of the original set.

A ⊆ B → (|A| ≤ |B| ∧ A is finite)

In particular, if A is a proper subset of B (that is, A ⊂ B), it strictly holds that |A| < |B|.

3) Finite union of finite sets: The union of two finite sets always produces another finite set. Furthermore, if the two sets are disjoint, the cardinality of their union equals the direct sum of their individual cardinalities.

A ∩ B = Ø → |A ∪ B| = |A| + |B|

4) Principle of inclusion-exclusion: When two finite sets share elements, the cardinality of their union is obtained by adding their individual sizes and subtracting the cardinality of their intersection to avoid double-counting common elements.

|A ∪ B| = |A| + |B| - |A ∩ B|

5) Intersection with a finite set: The intersection between a finite set and any other set (whether finite or infinite) always produces a finite set.

A is finite → A ∩ B is finite

This follows because the intersection A ∩ B is a subset of A, directly inheriting its finite property.

6) Difference of finite sets: When subtracting two sets where the minuend is finite, the result is always a finite set whose size does not exceed that of the starting set.

|A - B| = |A| - |A ∩ B|

Because only elements belonging to the original set A are removed, the resulting set retains a bounded number of elements.

7) Finite Cartesian product: The Cartesian product of two finite sets A and B produces a new finite set of ordered pairs. The resulting cardinality corresponds to the arithmetic product of the number of elements in each participating set.

|A × B| = |A| · |B|

8) Finiteness of the power set: If a set A is finite, its power set P(A) is also finite.

|P(A)| = 2^|A|

The cardinality of P(A) represents the total number of possible subsets that can be formed from the elements of A, always including the empty set and A itself.

Solved Practice Problems

To find the cardinality and verify whether a set is finite, the process involves either explicitly identifying its elements or applying the appropriate algebraic counting operations. Below, we solve four practical problems to reinforce these concepts.

Exercise 1

Given the set in set-builder notation D = {x ∈ Z⁺ | x is a divisor of 36 ∧ x is odd}, write its elements in roster form, determine whether it is a finite set, and find its cardinality.

Solution

We begin by listing all positive integer divisors of 36:

Div(36) = {1, 2, 3, 4, 6, 9, 12, 18, 36}

Next, we select only the values that satisfy the second condition, which requires them to be odd numbers:

D = {1, 3, 9}

Counting its members yields a countable and terminating collection. Therefore, D is a finite set with cardinality |D| = 3.

Exercise 2

Let A and B be two finite sets such that |A| = 14, |B| = 9, and |A ∪ B| = 18. Find the cardinality of the intersection |A ∩ B| and the cardinality of the relative complement |A − B|.

Solution

To calculate the number of shared elements, apply the principle of inclusion-exclusion for two finite sets:

|A ∪ B| = |A| + |B| − |A ∩ B|

Substitute the known values into the equation:

18 = 14 + 9 − |A ∩ B|

18 = 23 − |A ∩ B|

Solve for the intersection term:

|A ∩ B| = 23 − 18 = 5

With the shared elements determined, calculate the cardinality of the set difference |A − B|, which represents the elements that belong to A but not to B:

|A − B| = |A| − |A ∩ B| = 14 − 5 = 9

The intersection set contains 5 elements, and the relative complement contains 9 elements.

Exercise 3

Given the set of numbers A = {x ∈ Z | −1 ≤ x ≤ 2}, list its members in roster form, find its cardinality, and determine the number of elements in its power set P(A).

Solution

List the integer values that satisfy the given compound inequality:

A = {−1, 0, 1, 2}

Counting the elements confirms that its cardinality is |A| = 4.

To determine the number of elements in the power set P(A), use the power set cardinality formula based on the size of A:

|P(A)| = 2^|A|

Substitute the calculated cardinality:

|P(A)| = 2⁴ = 16

Thus, the power set P(A) is a finite collection consisting of 16 subsets.

Exercise 4

Consider the sets C = {x ∈ N | 1 ≤ x ≤ 5} and D = {x ∈ R | 1 ≤ x ≤ 5}. Determine whether each set is finite, justifying the answer based on the reference number system.

Solution

First, evaluate set C, whose universal set is the natural numbers. Between 1 and 5, there are a limited number of integers, which can be listed in roster form:

C = {1, 2, 3, 4, 5}

Because the counting process terminates and its size is |C| = 5, C is a finite set.

Next, examine set D, whose domain corresponds to the real numbers. The real number line has the property of density, meaning that between any two distinct real numbers there are infinitely many real numbers. Since it is impossible to enumerate its values through a one-to-one correspondence with a finite list, D is an infinite set, despite being numerically bounded between 1 and 5.