13 CLASSES OF SETS AND EXAMPLES - CULTURE VOCABULARY - 2024

The classes of sets can be classified into equal, finite and infinite, subsets, voids, disjoint or disjunctive, equivalent, unitary, superimposed or overlapping, congruent and non-congruent, among others.

A set is a collection of objects, but new terms and symbols are necessary to be able to speak sensibly about sets. For example, we say set of horses, set of real numbers, set of people, set of dogs, etc.

In ordinary language, the world in which we live is made sense by classifying things. Spanish has many words for such collections. For example, "a flock of birds," "a herd of cattle," "a swarm of bees," and "a colony of ants."

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

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 set 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 with 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 item appears in the list more than once, it is only counted 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. It is usually considered bad notation when an element is enumerated more than once.

2- Finite and infinite sets

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

• {Whole numbers between 2,000 and 2,005} = {2,001, 2,002, 2,003, 2,004}
• {Whole numbers 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 items could have been listed, but to save space, dots 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,…}
• {Whole numbers greater than 2,000} = {2,001, 2,002, 2,003, 2,004,…}

Both sets are infinite, since no matter how many items you try to enumerate, there are always more items in the set that cannot be listed, no matter how long you try. This time the dots '…' have a slightly different meaning, because they represent infinitely many unnumbered items.

3- Sets subsets

A subset is a part of a set.

• Example: Owls are a particular type of bird, so each 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 each element of S is an element of T. This is written as:

• S ⊂ T (Read "S is a subset of T")

The new symbol ⊂ means 'is a subset of'. So {owls} ⊂ {birds} because every 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 '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 does not fly.

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

Because 0 ∈ A, but 0 ∉ B, we read “0 belongs to set A”, but “0 does not belong to 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, no matter what X can 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 disjoints 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 set A is equal to the cardinal number of 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- Unit sets

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

For example:

• S = {a}
• Let B = {is an even prime number}

Therefore, B is a unit 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 constitute subsets of the universal set, which is named by the italicized capital letter U.

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

• U can be defined as the set of all living things on planet Earth. In that case, the set of all felines is a subset of U, the set of all fish is another subset of U.
• If U is defined as the set of all animals on planet earth, then the set of all felines is a subset of U, the set of all fish is another subset of U, but the set of all trees is not a subset of U.

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 overlapping sets.

10- Congruent sets.

They are those sets in which each element of A has the same distance relationship 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- Non-congruent sets

They are those in which the same distance relationship between each element in A cannot be established 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 non-congruent sets.

12- Homogeneous sets

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

• B {2, 8, 20, 100, 500}

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

13- Heterogeneous sets

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

• A {z, auto, π, buildings, block}

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

References

1. Brown, P. et al (2011). Sets and Venn diagrams. Melbourne, University of Melbourne.
2. Finite set. Recovered from: math.tutorvista.com.
3. Hoon, L. and Hoon, T (2009). Math Insights Secondary 5 Normal (Academic). Singapore, Pearson Education South Asia Pte Ld.
4. Recovered from: searchsecurity.techtarget.com.
5. Types of sets. Recovered from: math-only-math.com.