Types of Sets: Definition, Examples, and Symbols Explained

In Mathematics, sets are represented as the combination of objects whose elements are fixed and cannot be changed. A set is determined as a well-defined collection of objects. These objects are also known as elements of the set. The elements present in the set cannot be repeated in the set although can be written in any order. The set is denoted by capital letters.

Example: P = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Components of a set are embedded in curly brackets distributed by commas as can be seen in the above example.

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Different Types of Sets

The various types of sets with examples are given below:

• Null Set/Empty Set
• Singleton Set
• Finite Set
• Infinite Set
• Subset
• Power Set
• Universal Set
• Equivalent Sets
• Equal Sets
• Super Set

Types of sets in maths are important to learn not only to understand the theories in math but to also apply them in day-to-day life as arranging objects that belong to the alike category and keeping them in one group helps to find things easily and look clean as well.

Null set is a set that does not contain any element. The cardinality of the empty set is zero. The null set or the void set is expressed by the symbol ∅ and is read as phi. In roster form, ∅ is indicated by {}.

An empty set is said to be a finite set as the number of elements/symbols in an empty set is finite, i.e., zero(0).

Empty set or null set examples:

Example 1: P = {y : y is a leap year between 2004 and 2008}

As we can see, between 2004 and 2008, there was no leap year. Therefore, P = ϕ.

Example 2: R = {x : x denotes a whole number that is not a natural number given x ≠ 0}

As per the definition of whole numbers and natural numbers zero(0) is the only whole number that is not a natural number. If x ≠ 0, then there exists no other value possible for x. Hence, R = ϕ.

Example 3: Let P = {y : 3 < y < 4, y is a natural number}

Here P denotes an empty set because there does not lie any natural number between 3 and 4.

Singleton Set

A set that has only one element is termed a singleton set.

Singleton Set Examples:

• P= {y : y implies neither composite nor prime}

The given set P is a singleton set as it contains one element, i.e., one.

• Q = {y : y signifies a whole number that is less than one}

The set Q set includes only one element that is zero{0} hence it is an example of a singleton set.

Similarly;

• A = {p : p is a whole number that is not a natural number}

There is one whole number ‘zero’ which is not a natural number, hence it is an example of a singleton set.

• Set A = { 10 } is a singleton set.
• Set Y = {r : r is an even prime number}

Here Y is a singleton set because there exists only one prime number that is even and it is 2.

Finite Set

A set that contains a finite number of elements is named a finite set. In other words, we can say that a set that includes no element or a definite number of elements is said to be a finite set. The empty set is also termed a finite set.

Finite set example:

• Set P = {4,5,6,7,8,9,10} is a finite set, as it has a finite number of elements.
• The set of different colours in the rainbow is also an example of a finite set.

Similarly;

• M = {x : x ∈ M, x < 8}
• Q = {3, 5, 7, 11, 13, 17, 19 …… 113} are also examples of a finite set.

Learn about Cartesian Product of Sets

Infinite Set

Exactly opposite to the finite set, the infinite set will have an infinite number of elements. If a presented set is not finite, then it will be an infinite set.

OR

A set that has an infinite number of components is named an infinite set.

Infinite set example:

• A = {p : p is a whole number}

There are infinite whole numbers. Therefore, A is an infinite set.

• C= {z: z is the ordinate of a position on a provided line}

There can be infinite points on a line. Therefore, C is an infinite set.

Similarly;

• B = {x : x ∈ B, x > 2}
• D = {x : x ∈ D, x = 3m}
• Set of all prime numbers, Set of all even numbers, Set of all odd numbers are examples of an infinite set.
• All infinite sets cannot be represented in roster form.

Learn about Set Builder Notation

Subset

Consider A and B to be two sets. If each element of A is present in set B or we can say that if the elements of set A belong to set B, Then A is designated a subset of B and it is denoted by the notation A ⊆ B.

• The symbol ‘⊆’ is applied to signify ‘is a subset of’ or ‘is included in’.
• A ⊆ B; implies A is a subset of B in other words A is contained in B.
• B ⊆ A; implies B is a subset A.
• Every given set is a subset of itself.

Subset Examples:

Set A= {p, q, r, s, t, u}

Set B= {m, n, o, p, q, r, s, t, u}

Then we can state A ⊆ B.

Let us take another example; X = { 3, 4, 5, 6, 7, 8, 9 } and Y = { 6, 7 }. Hereabouts we can see that set Y is a subset of set X as all the components of set Y are in set X. Therefore, we can write Y ⊆ X.

Proper Subset: Consider A and B to be two sets. Then A is declared to be a proper subset of B if A is a subset of B and A is not equivalent to B. It is expressed as A ⊂ B.

OR

Any set say “P” is supposed to be a proper subset of ‘Q’ if there is at least one element in Q, which is not available in set P. That is, a proper subset is one that contains a few components of the original set.

Proper subset example:

If A = {2, 3, 4, 7, 8}

Here n(A) = 5

B = {1, 2, 3, 4, 7, 8, 10}

Here n(B) = 7

We can observe that all the elements of A are present in B but the element ‘1, 10’ of B is not available in A. Therefore, we say that A is a proper subset of B. Symbolically, this is written as A ⊂ B.

Note: No set is a proper subset of itself.

The null set denoted by the symbol ‘∅’ is a proper subset of every set.

Improper Subset: Suppose two sets, X and Y then X is an improper subset of Y if it includes all the elements of Y.

Learn about Cantor Set

Power Set

Let A be set, then the set of all the possible subsets of A is called the power set of A and is denoted by P(A). The number of components of the power set is given by 2n That is for a set A which covers n elements, the total number of subsets that can be created is 2n. From this, we can state that P(A) will have 2n elements.

Power Set Example 1:

For the set {x,y,z}:

• The empty set {} is a subset of {x,y,z}
• And these are subsets: {x}, {y} and {z}
• And these are subsets: {x,y}, {x,z} and {y,z}
• And {x,y,z} is actually a subset of {x,y,z} too.

And collectively they compose the Power Set:

P(S) = { {},{x}, {y}, {z}, {x,y}, {x,z}, {y,z}, {x,y,z} }

Example 2: If set A = {-2,3,6}, then obtain power set of A.

n=3, therefore 23number of elements will be present in the power set.

P(A)={{}, {-2}, {3}, {6}, {-2,3}, {3,6}, {6,-2}, {-2,3,6}}

Learn about Complement of a set

Universal Set

The universal set is normally indicated by U, and all its subsets by the letters A, B, C, etc. For example, for the set of all integers, the Universal Set can be the set of rational numbers, in human population studies, the universal set comprises all the people in the world.

OR

This is the set that is the foundation for every other set developed. Depending upon the circumstances, the universal set is chosen. It may be a finite or infinite set. All the other sets remain the subsets of the universal set.

Universal set example:

Consider if set A = {2,3,4}, set B = {4,5,6,7} and C = {6,7,8,9, 10}

Then, we will address the universal set as U = {2,3,4,5,6,7,8,9,10}

Note: As per the definition of the universal set, we can say that all the sets are subsets of the universal set.

Hence,

A ⊆ U

B ⊆ U

and C ⊆ U.

Learn about Finite and Infinite Sets

Equivalent Sets

The number of different elements in a given set A is termed as the cardinal number of A and is denoted by n(A).

If A {y : y ∈ N, x < 7}

The set A = {1, 2, 3, 4, 5, 6}

Therefore, n(A) = 6

Similarly;

P = set of letters in the word TESTBOOK.

P = {T, E, S, T, B, O, O, K}

Therefore, n(P) = 8.

Any two sets are stated to be equivalent sets if their cardinality is the same.

OR

Two sets P and Q are supposed to be equivalent if their cardinal number is identical, i.e., n(P) = n(Q). The symbol for expressing an equivalent set is ‘↔’.

Equivalent sets example:

• A = {3, 2, 5} Here n(A) = 3.
• B = {r, s, t} Here n(B) = 3.
• Therefore, A ↔ B.

Learn about Set Operations

Equal Sets

Any two sets are declared to be equal sets if and only if they are equivalent and as well as their elements are identical.

OR

Two sets P and Q are supposed to be equal if they hold the same elements. Each element of P is an element of Q and every element of Q is an element of P.

Equal sets example:

Let P{1, 2, 3, 4, 5} and Q={y : y , for 0<y<6 , y ∈ natural numbers}

• Writing Q in the tabular form {1, 2, 3, 4, 5}
• Here, each element of P is an element of Q, i.e., P ⊆ Q.
• Also, every element of Q is an element of P, i.e., Q ⊆ P.
• Therefore, sets P and Q stand for equal sets.

Learn about Roster Notation

Super Set

Whenever a given set P is a subset of set Q, we say the Q is a superset of P and we address it as Q ⊇ P. The symbol ⊇ is applied to denote ‘is a superset of’.

For example;

A = {a, e, i, o, u}

B = {a, b, c, d, e, f…………., z}

Here A ⊆ B i.e., A is a subset of B however B ⊇ A i.e., B is a superset of A

Learn about Union of Sets

Formulas for Sets

1. n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

→ Number of elements in the union of sets A and B.

2. n(A ∩ B) = n(A) + n(B) − n(A ∪ B)

→ Number of elements in the intersection of sets A and B.

3. n(A') = n(U) − n(A)

→ Number of elements in the complement of set A (U is the universal set).

4. n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(B ∩ C) − n(C ∩ A) + n(A ∩ B ∩ C)

→ Formula for union of three sets with all intersections considered.

5. If A ⊆ B, then:

- A ∪ B = B

- A ∩ B = A

→ Properties of subsets.

6. A ∪ ∅ = A

→ Union of any set with the empty set is the set itself.

7. A ∩ ∅ = ∅

→ Intersection of any set with the empty set is the empty set.

8. A ∪ A = A

→ Union of a set with itself is the same set.

9. A ∩ A = A

→ Intersection of a set with itself is the same set.

10. A ∪ A' = U

→ Union of a set and its complement gives the universal set.

11. A ∩ A' = ∅

→ Intersection of a set and its complement is the empty set.

Sets Symbols

Set symbols are used to define the elements of a given set. The table presents some of these symbols with their meaning.

Symbols	Meaning
U	Universal set
n(Y)	Cardinal number of set Y
c ∈ P	‘c’ is an element of set P
a ∉ Q	‘a’ is not an element of set Q
∅	Null or empty set
{}	Denotes a set
X ⊆ Y	Set X is a subset of set Y
Y ⊇ X	Set Y is the superset of set X

Properties of Types of Sets

1. Empty Set (Null Set)

• It contains no elements.
• Symbol: ∅ or { }.
• It is a subset of every set.
• It has only one subset: itself.

2. Singleton Set

• A set that contains only one element.
• Example: {7} or {apple}.
• It has two subsets: the empty set and the set itself.

3. Finite Set

• Contains a countable number of elements.
• You can list all the elements.
• Example: {2, 4, 6, 8}.

4. Infinite Set

• Contains uncountable or endless elements.
• Cannot list all elements.
• Example: Set of all natural numbers N = {1, 2, 3, ...}.

5. Equal Sets

• Two sets are equal if they have exactly the same elements.
• Order doesn’t matter: {1, 2} = {2, 1}.
• Symbols: A = B.

6. Equivalent Sets

• Sets that have the same number of elements, but the elements may be different.
• Example: {a, b, c} and {1, 2, 3} are equivalent.
• Symbol: A ~ B.

7. Subset

• A set A is a subset of set B if every element of A is also in B.
• Symbol: A ⊆ B.
• The empty set is a subset of every set.

8. Power Set

• The set of all subsets of a given set.
• If a set has n elements, the power set has 2ⁿ elements.

9. Universal Set

• The set that contains all possible elements for a given context.
• Usually represented by the symbol U.
• Every set is a subset of the universal set.

10. Disjoint Sets

• Two sets are disjoint if they have no common elements.
• Example: A = {1, 2}, B = {3, 4} → A ∩ B = ∅

Operations of Sets

1. Union of Sets (A ∪ B)

• The union of two sets A and B includes all elements that are in A, or in B, or in both.
• Symbol: ∪
• Example:
  A = {1, 2, 3}, B = {3, 4, 5}
  A ∪ B = {1, 2, 3, 4, 5}

2. Intersection of Sets (A ∩ B)

• The intersection includes only the elements that are common to both A and B.
• Symbol: ∩
• Example:
  A = {1, 2, 3}, B = {2, 3, 4}
  A ∩ B = {2, 3}

3. Difference of Sets (A − B)

• The difference A − B contains elements that are in A but not in B.
• Symbol: −
• Example:
  A = {1, 2, 3}, B = {2, 3, 4}
  A − B = {1}
  (Note: B − A = {4})

4. Complement of a Set (A')

• The complement of A contains all elements in the universal set U that are not in A.
• Symbol: A' or Aᶜ
• Example:
  U = {1, 2, 3, 4, 5}, A = {2, 3}
  A' = {1, 4, 5}

5. Symmetric Difference (A Δ B)

• The symmetric difference includes elements in A or B but not in both.
• Symbol: Δ
• Example:
  A = {1, 2, 3}, B = {3, 4, 5}
  A Δ B = {1, 2, 4, 5}

Types of Sets Examples

Example 1:
Let’s say we have two sets:
Set P = {M, A, T, H}
Set Q = {H, A, T, S}

Are these sets the same or different?
To check, we look at the elements in each set. Set P has M, A, T, H. Set Q has H, A, T, S. Since P has M and Q has S, they are not the same.

Answer: Sets P and Q are different sets.

Example 2:
Consider these sets:
Set M = {2, 4, 6}
Set N = {a, e, i, o, u}
Set O = {u, o, i, e, a}

Are any of these sets related?

• Set M has 3 elements.
• Set N and O both have 5 elements.

When two sets have the same number of elements, they are called equivalent sets. So, M and N are not equivalent because they have different counts, but N and O both have 5 elements, so N and O are equivalent sets.

Also, when two sets have the exact same elements, they are called equal sets. Since N and O contain the same letters just in different order, N and O are equal sets.

Answer:

• M and N are not equivalent.
• N and O are both equivalent and equal sets.

Example 3:
Let’s take these sets:
Set R = {x, y, z}
Set S = {w, x, y, z}

Find:
a) The union of R and S (written as R ∪ S) — this means all elements in either set.
b) The intersection of R and S (written as R ∩ S) — this means elements common to both sets.

Solution:
a) R ∪ S = {w, x, y, z} (all elements from both sets, no repeats)
b) R ∩ S = {x, y, z} (elements both sets share)

Answer:
a) R ∪ S = {w, x, y, z}
b) R ∩ S = {x, y, z}

If you are checking Types of Sets article, also check related maths articles:
Intersection of sets	Basic set theory
Set Theory Symbols	Set Difference
Union of Sets	Cartesian Product of Sets

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