SET THEORY: CHARACTERISTICS, ELEMENTS, EXAMPLES, EXERCISES - MATHS - 2025

The set theory is a branch of mathematical logic-which is responsible for the study of relationships between entities called sets. The sets are characterized by being collections of objects of the same nature. Said objects are the elements of the set and can be: numbers, letters, geometric figures, words that represent objects, the objects themselves and others.

It was Georg Cantor, towards the end of the 19th century, who proposed set theory. While other notable mathematicians in the 20th century made their formalization: Gottlob Frege, Ernst Zermelo, Bertrand Russell, Adolf Fraenkel among others.

Figure 1. Venn diagram of the sets A, B and their intersection A⋂ B. (Own elaboration).

Venn diagrams are the graphic way to represent a set, and it consists of a closed plane figure within which are the elements of the set.

For example, in figure 1 two sets A and B are shown, which have elements in common, the elements common to A and B. These form a new set called the intersection set of A and B, which is written in the form symbolic as follows:

A ∩ B

characteristics

The set is a primitive concept as it is in geometry the concept of point, line or plane. There is no better way to express the concept than by pointing out examples:

Set E formed by the colors of the flag of Spain. This way of expressing the set is called by comprehension. The same set E written by extension is:

E = {red, yellow}

In this case, red and yellow are elements of set E. It should be noted that the elements are listed in braces and are not repeated. In the case of the Spanish flag, there are three colored stripes (red, yellow, red), two of which are repeated, but the elements are not repeated when the whole is expressed.

Suppose the set V formed by the first three vowel letters:

V = {a, e, i}

The power set of V, denoted by P (V) is the set of all sets that can be formed with the elements of V:

P (V) = {{a}, {e}, {i}, {a, e}, {a, i}, {e, i}, {a, e, i}}

Types of sets

Finite set

It is a set in which its elements are countable. Examples of finite sets are the letters of the Spanish alphabet, the vowels of Spanish, the planets of the Solar system, among others. The number of elements in a finite set is called its cardinality.

Infinite set

An infinite set is understood to be all that the number of its elements is uncountable, since no matter how large the number of its elements may be it is always possible to find more elements.

An example of an infinite set is the set of natural numbers N, which in extensive form is expressed as follows:

N = {1, 2, 3, 4, 5,….} Is clearly an infinite set, since no matter how large a natural number may be, the next largest can always be found, in an endless process. Clearly the cardinality of an infinite set is ∞.

Empty set

It is the set that does not contain any element. The empty set V is denoted by Ø or by a pair of keys without elements inside:

V = {} = Ø.

The empty set is unique, therefore it must be incorrect to say "an empty set", the correct form is to say "the empty set".

Among the properties of the empty set we have that it is a subset of any set:

Ø ⊂ A

Furthermore, if a set is a subset of the empty set, then necessarily said set will be the vacuum:

A ⊂ Ø ⇔ A = Ø

Unitary set

A unit set is any set that contains a single element. For example, the set of natural satellites of the Earth is a unitary set, whose only element is the Moon. The set B of integers less than 2 and greater than zero only has element 1, therefore it is a unit set.

Binary set

A set is binary if it only has two elements. For example the set X, such that x is a real number solution of x ^ 2 = 2. This set by extension is written like this:

X = {-√2, + √2}

Universal set

The universal set is a set that contains other sets of the same type or nature. For example, the universal set of natural numbers is the set of real numbers. But the real numbers is a universal set also of the whole numbers and the rational numbers.

Core items

- Relations between sets

In assemblies, various types of relationship can be established between them and their elements. If two sets A and B have exactly the same elements between them, an equality relationship is established, denoted as follows:

A = B

If all the elements of a set A belong to a set B, but not all the elements of B belong to A, then between these sets there is an inclusion relation that is denoted like this:

A ⊂ B, but B ⊄ A

The above expression reads: A is a subset of B, but B is not a subset of A.

To indicate that some element or elements belong to a set, the membership symbol ∈ is used, for example to say that x element or elements belong to the set A is written symbolically like this:

x ∈ A

If an element does not belong to the set A, this relation is written like this:

and ∉ A

The membership relationship exists between the elements of a set and the set, with the sole exception of the power set, the power set being the collection or set of all possible sets that can be formed with the elements of said set.

Suppose V = {a, e, i}, its power set is P (V) = {{a}, {e}, {i}, {a, e}, {a, i}, {e, i}, {a, e, i}}, in that case the set V becomes an element of the set P (V) and can be written:

V ∈ P (V)

- Properties of inclusion

The first property of inclusion establishes that every set is contained in itself, or in other words, that it is a subset of itself:

A ⊂ A

The other property of inclusion is transitivity: if A is a subset of B and B is in turn a subset of C, then A is a subset of C. In symbolic form, the transitivity relation is written as follows:

(A ⊂ B) ^ (B ⊂ C) => A ⊂ C

Below is the Venn diagram corresponding to the transitivity of inclusion:

Figure 2. (A ⊂ B) ^ (B ⊂ C) => A ⊂ C

- Operations between sets

Intersection

The intersection is an operation between two sets that gives rise to a new set belonging to the same universal set as the first two. In that sense, it is a closed operation.

Symbolically the intersection operation is formulated like this:

A⋂B = {x / x∈A ^ x∈B}

An example is the following: the set A of the letters of in the word “elements” and the set B of the letters of the word “repeated”, the intersection between A and B is written like this:

A⋂B = {e, l, m, n, t, s} ⋂ {r, e, p, t, i, d, o, s} = {e, t, s}. The universal set U of A, of B and also of A⋂B is the set of the letters of the Spanish alphabet.

Union

The union of two sets is the set formed by the elements common to the two sets and the non-common elements of the two sets. The union operation between sets is expressed symbolically like this:

A∪B = {x / x∈A vx∈B}

Difference

The difference operation of set A minus set B is denoted by AB. AB is a new set formed by all the elements that are in A and that do not belong to B. Symbolically it is written like this:

A - B = {x / x ∈ A ^ x ∉ B}

Figure 3. A - B = {x / x ∈ A ^ x ∉ B}

Symmetric difference

The symmetric difference is an operation between two sets where the resulting set is made up of the elements not common to the two sets. The symmetric difference is symbolically represented like this:

A⊕B = {x / x∈ (AB) ^ x∈ (BA)}

Examples

Example 1

The Venn diagram is a graphical way of representing sets. For example, the set C of the letters in the word set is represented like this:

Example 2

It is shown below by Venn diagrams that the set of vowels in the word "set" is a subset of the set of letters in the word "set".

Example 3

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.

Example 4

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.

Example 5

Given the sets A = {2, 4, 6, 8} and B = {1, 2, 4, 7, 9}, determine AB and BA.

A - B are the elements of A that are not in B:

A - B = {6, 8}

B - A are the elements of B that are not in A:

B - A = {1, 7, 9}

Solved exercises

Exercise 1

Write in symbolic form and also by extension the set P of even natural numbers less than 10.

Solution: P = {x∈ N / x <10 ^ x mod 2 = 0}

P = {2, 4, 6, 8}

Exercise 2

Suppose the set A that formed by the natural numbers that are factors of 210, and the set B that formed by the prime natural numbers less than 9. Determine by extension both sets and establish what relationship there is between the two sets.

Solution: To determine the elements of set A, we must begin by finding the factors of the natural number 210:

210 = 2 * 3 * 5 * 7

Then the set A is written:

A = {2, 3, 5, 7}

We now consider the set B, which is the primes less than 9. 1 is not prime because it does not meet the definition of prime: "a number is prime if and only if it has exactly two divisors, 1 and the number itself." The 2 is even and at the same time it is prime because it meets the definition of a prime, the other primes less than 9 are 3, 5 and 7. So the set B is:

B = {2, 3, 5, 7}

Therefore the two sets are equal: A = B.

Exercise 3

Determine the set whose elements x are different from x.

Solution: C = {x / x ≠ x}

Since every element, number or object is equal to itself, the set C cannot be other than the empty set:

C = Ø

Exercise 4

Let the set of N's of natural numbers and Z be the set of integers. Determine N ⋂ Z and N ∪ Z.

Solution:

N ⋂ Z = {x ∈ Z / x ≤ 0} = (-∞, 0]

N ∪ Z = Z because N ⊂ Z.

References

1. Garo, M. (2014). Mathematics: quadratic equations: How solve a quadratic equation. Marilù Garo.
2. Haeussler, EF, & Paul, RS (2003). Mathematics for management and economics. Pearson Education.
3. Jiménez, J., Rodríguez, M., Estrada, R. (2005). Mathematics 1 SEP. Threshold.
4. Preciado, CT (2005). Mathematics Course 3rd. Editorial Progreso.
5. Mathematics 10 (2018). "Examples of Finite Sets". Recovered from: matematicas10.net
6. Wikipedia. Set theory. Recovered from: es.wikipedia.com