# The 14 types of assemblies: ways to classify elements

Human beings like to classify the world. Since classical times, in Ancient Greece, great philosophers like Aristotle elaborated complex classification systems for plants, animals and other elements that make up reality

In the modern world we have provided ourselves with sciences such as mathematics and logic in order 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 will see what the different classes of sets are** , as well as detailing how they are expressed by giving examples.

## What is an ensemble?

This is **a grouping of elements that are in the same category or share a typology** . Each of its elements is differentiated from the others.

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 keys in which the elements of the set are placed.

Thus, **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, the expression of the sets can list all the elements that make them up (examples A,B and C) or simply put a sentence that defines everything that makes them up (examples D and E).

**When writing a set it is necessary to be clear and the definition should not be misleading** . For example, the set {beautiful paintings} is not a good set, since defining what is meant by beautiful art is totally 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 **if they contain the same elements** .

For example:

A = {odd numbers 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 in 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 the same element (e.g. B {1,1,3,5…}) is repeated in a set **the repetition should be ignored, as it may be due to an error** in the notation.

### 2. Finite sets

Finite sets are those in which **it is possible to count all their elements** .

{pair numbers from 2 to 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 points ‘…’:

{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 elements** :

{even numbers} = {2,4,6,8,10…}

In this example you can list hundreds of items, but you will never reach the end. In this case the three points do not represent concrete values, but continuity.

### 4. Sub-assemblies

As the name implies, **are sets within sets with more elements** .

For example, the ulna is a bone of the human body, so we would say that the ulna bone set is a subset of the bone set. So:

C = {cubitus bones} and H = {human bones}, then C ⊂ H.

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

To represent the opposite, i.e. that a set is not a subset of another, the symbol ⊄ is used.

{arachnids} ⊄ {insects}

Spiders, although arthropods, do not fall into the category of insects.

**To represent the relationship of a given element to a set we use the symbol ∈** , which reads ‘element of’.

Returning to the previous example, a spider is an element that constitutes the category arachnids, so spider ∈ arachnids, on the other hand, 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 has no 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. Disjunctive sets

Two sets **are disjunctive if they do not share elements at all** .

P = {dog breeds} and G = {cat breeds}.

These are part of the most frequent types of sets, as they go very well to classify in a clear and orderly way.

### 7. Equivalent sets

Two sets are equivalent if **have the same number of elements, but they are not the same** . For example:

A = {1,2,3} and B = {A,B,C}

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

### 8. Unit sets

They are sets in which there is only one element:

A = {1}

### 9. Universal or reference set

A set is universal **if it is made up 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 letter U in italics.

For example, U can be defined as all the living beings on the planet. So, animals, plants and fungi would be three subsets within U.

If, for example, we consider that U is 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, using 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

They are two sets whose elements **have the same distance between them** . They are usually of the numerical or alphabetical type. For example:

A = {1,2,3,4,…} and B = {10,11,12,13,14,…}

These two sets are congruent, since their elements have the same distance between them, being a 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 **their elements do not have the same distance between them** .

A = {1,2,3,4,5,…} and B = {1,3,5,7,9,…}

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 congruent sets.

A non-congruent set separately 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** , for example:

C = {1,3,7,11,21,93}

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

### 13. Homogeneous

All the elements of the set **belong to the same category, i.e. 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 not constitute a clear category by itself, but the inclusion of its elements **seems to be due to chance** :

A = {5, plane, X, chaos}

#### Bibliographic references:

- Brown, P. et al (2011). Sets and Venn diagrams. Melbourne, University of Melbourne.
- “Types of assemblies” (undated). In There are Types. Available at: https://haytipos.com/conjuntos/ [Accessed: 21 August 2019].
- Types of sets (undated). Retrieved from: math-only-math.com.