**What are set types?**

The **types of sets** They are all those ways of grouping elements that may or may not have characteristics in common. Sets can be classified as equal, finite and infinite, subsets, empty, disjoint or disjunctive, equivalent, unitary, superimposed or overlapping, congruent and non-congruent, among others.

A set is a grouping of objects of the same category, or that share features, typologies or characteristics in common. For example, we say a set of horses, a set of real numbers, a set of people, a set of dogs, etc.

In mathematics, something similar is done when numbers, geometric figures, etc. are classified. The objects in these sets are called elements of the set.

**description of a set**

A set can be described by listing all its elements. For example,

S = {1, 3, 5, 7, 9}.

«S is the set whose elements are 1, 3, 5, 7 and 9». The five elements of the array are separated by commas and are listed in braces.

A set can also be delimited by presenting a definition of its elements in square brackets. Thus, the set S above can also be written as:

S = {odd integers less than 10}.

A set must be well defined. This means that the description of the elements of a set must be clear and unambiguous.

For example, {tall people} is not a set, because people tend to disagree about what ‘tall’ means. An example of a well-defined set is

T = {letters of the alphabet}.

**Types of sets**

**1. Equal sets**

Two sets are equal if they have exactly the same elements.

For example:

– If A = {Vowels of the alphabet} and B = {a, e, i, o, u} it is said that A = B.

– On the other hand, the sets {1, 3, 5} and {1, 2, 3} are not the same, because they have different elements. This is written as {1, 3, 5} ≠ {1, 2, 3}.

– The order in which the elements are written inside the brackets does not matter at all. For example, {1, 3, 5, 7, 9} = {3, 9, 7, 5, 1} = {5, 9, 1, 3, 7}.

– If an element appears in the list more than once, it is counted only once. For example, {a, a, b} = {a, b}.

The set {a, a, b} has only the two elements a and b. The second mention of a is unnecessary repetition and can be ignored. Normally, bad notation is considered when an element is listed more than once.

**2. Finite and infinite sets**

Finite sets are those in which all the elements of the set can be counted or enumerated. Here are two examples:

– {Integers between 2,000 and 2,005} = {2,001, 2,002, 2,003, 2,004}

– {Integers between 2,000 and 3,000} = {2,001, 2,002, 2,003, …, 2,999}

The three dots ‘…’ in the second example represent the other 995 numbers in the set. All elements could have been listed, but to save space, periods were used instead.

This notation can only be used if it is completely clear what it means, as in this situation.

A set can also be infinite – all that matters is that it is well defined. Here are two examples of infinite sets:

– {Even numbers and integers greater than or equal to two} = {2, 4, 6, 8, 10, …}

– {Integers greater than 2,000} = {2,001, 2,002, 2,003, 2,004, …}

Both sets are infinite, since no matter how many elements you try to enumerate, there are always more elements in the set that cannot be enumerated, no matter how long you try.

This time the dots ‘…’ have a slightly different meaning, because they represent infinitely many unenumerated items.

**3. Sets subsets**

A subset is a part of a set.

– Example: Owls are a particular kind of bird, so every owl is also a bird. In the language of sets, it is expressed by saying that the set of owls is a subset of the set of birds.

A set S is called a subset of another set T, if every element of S is an element of T. This is written as:

– S ⊂ T (Read “S is a subset of T”)

The symbol ⊂ means ‘is a subset of’. So {owls} ⊂ {birds} because each owl is a bird.

– If A = {2, 4, 6} and B = {0, 1, 2, 3, 4, 5, 6}, then A ⊂ B,

Because every element of A is an element of B.

The symbol ⊄ means ‘is not a subset’.

This means that at least one element of S is not an element of T. For example:

– {Birds} ⊄ {flying creatures}

Because an ostrich is a bird, but it doesn’t fly.

– If A = {0, 1, 2, 3, 4} and B = {2, 3, 4, 5, 6}, then A ⊄ B

Because 0 ∈ A, but 0 ∉ B, one reads “0 belongs to the set A”, but “0 does not belong to the set B”.

**4. Empty set**

The symbol Ø represents the empty set, which is the set that has no elements at all. Nothing in the entire universe is an element of Ø:

– | Ø | = 0 and X ∉ Ø, it doesn’t matter what X might be.

There is only one empty set, because two empty sets have exactly the same elements, so they must be equal to each other.

**5. Disjoint or disjunctive sets**

Two sets are called disjoint if they have no elements in common. For example:

– The sets S = {2, 4, 6, 8} and T = {1, 3, 5, 7} are disjoint.

**6. Equivalent sets**

It is said that A and B are equivalent if they have the same number of elements that constitute them, that is, the cardinal number of the set A is equal to the cardinal number of the set B, n (A) = n (B). The symbol to denote an equivalent set is ‘↔’.

– For example:

A = {1, 2, 3}, therefore n(A) = 3

B = {p, q, r}, therefore n(B) = 3

Therefore, A ↔ B

**7. Unitary sets**

It is a set that has exactly one element in it. In other words, there is only one element that makes up the set.

For example:

–S = {a}

– Let B = {is an even prime number}

Therefore, B is a unitary set because there is only one prime number that is even, that is, 2.

**8. Universal or referential set**

A universal set is the collection of all objects in a particular context or theory. All other sets in that frame are subsets of the universal set, which is named by the capital italic letter *OR*.

The precise definition of *OR* it depends on the context or theory under consideration. For example:

– It can be defined *OR* as the set of all living things on planet Earth. In that case, the set of all cats is a subset of *OR*the set of all fish is another subset of *OR*.

– If defined *OR* as the set of all animals on planet Earth, then the set of all felines is a subset of *OR*the set of all fish is another subset of *OR*but the set of all trees is not a subset of *OR*.

**9. Overlapping or overlapping sets**

Two sets that have at least one element in common are called overlapping sets.

– Example: Let X = {1, 2, 3} and Y = {3, 4, 5}

The two sets X and Y have one element in common, the number 3. Therefore, they are called superposed sets.

**10. Congruent sets**

They are those sets in which each element of A has the same distance relation with its image elements of B. Example:

– B {2, 3, 4, 5, 6} and A {1, 2, 3, 4, 5}

The distance between: 2 and 1, 3 and 2, 4 and 3, 5 and 4, 6 and 5 is one (1) unit, so A and B are congruent sets.

**11. Incongruent sets**

They are those in which the same distance relationship cannot be established between each element of A with its image in B. Example:

– B {2, 8, 20, 100, 500} and A {1, 2, 3, 4, 5}

The distance between: 2 and 1, 8 and 2, 20 and 3, 100 and 4, 500 and 5 is different, so A and B are incongruent sets.

**12. Homogeneous sets**

All the elements that make up the set belong to the same category, genus or class. They are of the same type. Example:

–B {2, 8, 20, 100, 500}

All the elements of B are number, so the set is considered homogeneous.

**13. Heterogeneous sets**

The elements that are part of the set belong to different categories. Example:

– A {z, car, π, buildings, block}

There is no category to which all the elements of the set belong, therefore, it is a heterogeneous set.

**References**

Brown, P. et al (2011). Sets and Venn diagrams. Melbourne, University of Melbourne.

Finite set. Retrieved from math.tutorvista.com.