# The 14 classes of sets: ways of classifying elements

**Let's see what the different classes of sets consist of: categories that encompass elements.**

Human beings like to classify the world. As far back as classical times, in Ancient Greece, great philosophers such as Aristotle elaborated complex classificatory systems for plants, animals and other elements that make up reality.

In the modern world we have been provided with sciences such as mathematics and logic to be able to express in an objective and numerical way concepts proper to philosophy.

Sets are collections of different elements, which are expressed by means of numerical expressions. In this article **we are going to see what the different classes of sets are**in addition to detailing in depth how they are expressed by giving examples.

## What is a set?

It is **a grouping of elements that are within the same category or share a common typology.**. Each of its elements are differentiated from one another.

In mathematics and other sciences, sets are represented numerically or symbolically, and are named with a letter of the alphabet followed by the symbol '=' and braces in which the elements of the set are placed inside.

Thus, a set can be represented as follows **a set can be represented in the following ways**:

- A = {1,2,3,4,5}
- B ={blue, green, yellow, red}
- C ={pink, daisy, geranium, sunflower}
- D = {even numbers}
- E = {consonants of the Latin alphabet}

As can be seen in these examples, in the expression of sets we can list all the elements that compose it (examples A, B and C) or, simply, put a phrase that defines everything that constitutes it (examples D and E).

**When writing a set, it is necessary to be clear and that the definition is not misleading.**. For example, the set {beautiful pictures} is not a good set, since defining what is meant by beautiful art is entirely subjective.

## Types of sets, and examples

In total there are about 14 different types of sets, useful for mathematics and philosophy.

### 1. Equal sets

Two sets are equal **In the case that they contain the same elements.**.

For example: A = {odd numbers from 1 to 15} and B = {1,3,5,7,9,11,13,15}, then A = B.

If two sets do not have the same elements and are therefore not equal, their inequality is represented by the symbol '≠'. C = {1,2,3} and D = {2,3,4}, therefore C ≠ D.

The order of the elements of both sets does not matter, as long as they are the same. E = {1,4,9} and F = {4,9,1}, therefore E = F.

If in a set the same element is repeated (e.g., B {1,1,3,5...}) **the repetition should be ignored, since it is possible that it is due to an error.** in the annotation.

### 2. Finite sets

Finite sets are those in which **it is possible to count all their elements**. {even numbers from 2 to 10} = {2,4,6,8,10} = {2,4,6,8,10}.

When in a set there are many elements but these are concrete and it is clear what they are, they are represented by three dots '...': {odd numbers from 1001 to 1501} = {1001,1003,1005,...,1501}

### 3. Infinite sets

This is the opposite of finite sets. **In infinite sets there are infinite number of elements**: {even numbers} = {2,4,6,8,10...}

In this example you can list hundreds of elements, but you will never get to the end. In this case the three dots do not represent concrete values, but continuity.

### 4. Subsets

As the name suggests **these are sets within sets with more elements.**.

For example, the ulna is a bone of the human body, for this reason we would say that the set of ulnar bones is a subset of the set of bones. Thus: C = {ulnar bones} and H = {human bones}, then C ⊂ H.

This expression above is read as C is a subset of H.

To represent the converse, i.e., that one set is not a subset of another, the symbol ⊄ is used. {arthropods} ⊄ {insects}

Spiders, although arthropods, are not within the insect category.

**To represent the relationship of a given element to a set we use the symbol ∈**which is read 'element of'.

Returning to the previous example, a spider is an element that constitutes the category arachnids, so spider ∈ arachnids, on the other hand, it is not part of the category insects, so spider ∉ insects.

- You may be interested in : "The 6 levels of ecological organization (and their characteristics)".

### 5. Empty set

**This is a set that does not have any elements.**. It is represented by the symbol Ø or by two empty keys {} and, as can be deduced, no element of the universe can constitute this set, since if it does, it automatically ceases to be an empty set. | Ø | = 0 and X ∉ Ø, no matter what X may be.

### 6. Disjoint or disjunctive sets

Two sets **are disjunctive if they do not share elements at all.**. P = {breeds of dogs} and G = {breeds of cats}.

These are part of the most frequent classes of sets, since they go very well to classify in a clear and orderly manner.

### 7. Equivalent sets

Two sets are equivalent if **they have the same number of elements, but without these being the same**. For example: A = {1,2,3} and B = {A,B,C}

Thus,n (A) = 3, n (B) = 3. Both sets have exactly three elements, which means that they are equivalent. This is represented as follows: A ↔️ B.

### 8. Unitary sets

These are sets in which there is only one element: A = {1}

### 9. Universal or referential set

A set is universal **if it consists of all the elements of a particular context or a particular theory.**. All the sets in this framework are the subsets of the universal set in question, which is represented by the italicized letter U.

For example, U can be defined as the set of all living things on the planet. Thus, animals, plants and fungi would be three subsets within U.

If, for example, we consider U to be all the animals on the planet, subsets of it would be cats and dogs, but not plants....

### 10. Overlapping or overlapping sets

These are two or more sets that **share at least one element**. They can be represented visually, by means of Venn diagrams. For example. A = {1,2,3} and B = {2,4,6}.

These two sets have in common the number 2.

### 11. Congruent sets

These are two sets whose elements **have the same distance between them**. They are usually of numerical or alphabetical type. For example: A = {1,2,3,3,4,...} and B = {10,11,12,13,13,14,...}

These two sets are congruent, since their elements have the same distance between them, being one unit of difference in each link of the sequence.

### 12. Non-congruent sets.

Contrary to the previous point, non-congruent sets are those in which the elements do not have the same distance between them. **their elements do not have the same distance between them**. A = {1,2,3,4,5,...} and B = {1,3,5,7,7,9,...}

In this case it can be seen that the elements of each set have different distances, being a distance of one unit in set A and a distance of two in set B. Therefore, A and B are not sets congruent to each other.

A separately noncongruent set is one in which it is not possible to establish a clear formula or pattern to explain why it has the elements that constitute it. **it is not possible to establish a clear formula or pattern to explain why it has the elements that make it up**for example: C = {1,3,7,11,21,93}

In this case it is not possible to know by way of mathematics why this set has these numbers.

### 13. Homogeneous

All the elements of the set **belong to the same category, that is to say, they are of the same type**: A = {1,2,3,4,5} B ={blue,green,yellow,red} C ={a,b,c,d,el}

### 14. Heterogeneous

The elements of the do not constitute a clear category by themselves, but rather the inclusion of their elements **seems to be due to chance**: A = {5, plane, X, chaos}

Bibliographical references:

- Brown, P. et al (2011). Sets and Venn diagrams. Melbourne, University of Melbourne.
- "Types of sets" (n/f.). In There are Types. Available from: https://haytipos.com/conjuntos/ [Accessed: 21 August 2019].
- Types of sets (s/f). Recuperado de: math-only-math.com.

(Updated at Apr 13 / 2024)