Superset is one of the not-so-common topics in the set theory, as this is not used as much as its related term i.e., Subset. A superset is a set that contains all of the items of another set, known as the subset. We know that if B is contained within A which means A contains B. In other words, if B is a subset of A, then A is its superset.

In this article, the concept of superset is discussed in plenty of detail. Other than that, its definition, symbols, properties, and several solved examples as well.

What is a Superset?

If we have two sets, a superset in Maths is a set that includes almost all of the items of the smaller set. If P and Q are two sets, and P is the superset of Q, then Q is the smaller set, and all of Q’s elements are present in P. Components/members of a given set are entities or items that belong to a certain sort of set. In arithmetic, relations, and functions, a set is commonly represented by capital letters, whereas the components are represented by lowercase letters. All of the items are enclosed by braces'{}’.

Read more about Set Theory.

Superset Definition

Set A is termed the superset of set B if all of the components of set B are also elements of set A. For example, if set A = {21, 22, 23, 24} and set B = {21, 23, 24} we may say that set A is the superset of B. Because the components of B [(i.e.,)21, 23, 24] are in set A. We may also state that B is not a superset of A.

The following illustration shows the relationship between the set and its superset using Venn Diagram.

Superset Symbol

The superset symbol, often known as or, is a mathematical symbol that represents the notion of one set being a superset of another. A set A is said to be a superset of another set B if every constituent of set B is also an element of set A. In other words, set A contains all of the components in set B.

Here’s what the symbols ⊃ and ⊇ stand for:

1. ⊃ (Superset Symbol): This symbol represents a strictly superset connection. If A ⊃ B, then set A is a superset of set B and must contain at least one element not found in set B. In other words, A is larger than B and contains items not present in B.
2. ⊇ (Superset or Equal Symbol): This symbol represents a superset connection that may or may not be equal. If A ⊇ B, it signifies that set A is a superset of set B, and it may include all of the same items as set B as well as more elements.

These connections can be expressed mathematically as follows:

1. A ⊃ B => This can be translated as “A is a strict superset of B” or “A contains all of B’s elements and possibly more.”
2. A ⊇ B => This is equivalent to saying “A is a superset of B, including the possibility that they are equal.”

The superset symbol is widely used in mathematics and set theory to express connections between sets, and it is critical to understanding set inclusion and confinement.

Superset Example

Let Y = {21, 22, 23, 24, 25, 26} and X = {21, 22, 23, 25, 26}

In the two sets above, every element of X is also an element of Y, and the number of elements of X is smaller than the number of elements of Y.

In other words, n(x) = 4 and n(Y) = 6

⇒ n(x) < n(Y)

As a result, Y is the superset of X.

Other, than this example we can give all the general sets as as supersets of each other as follows: 

N ⊃ W ⊃ Z ⊃ Q ⊃ R ⊃ C

Where,

• N is the set of Natural Numbers, 
• W is the set of Whole Numbers,
• Z is the set of Integers,
• Q is the set of Rational Numbers,
• R is the set of Real Numbers, and 
• C is the set of Complex Numbers.

Learn more about Subset of Real Numbers.

Proper and Improper Superset

A correct superset is often referred to as a stringent superset. If set X is the correct superset of set W, then all of set W’s elements are in X, but set X must have at least one member that is not in set W.

Take, for example, four sets.

W = {u, v, w}

X = {u, v, w, x}

Y = (u, v, w}

Z = {u, v, y}

Because X is not equal to W, it is the proper superset of W from the sets described above.

Y is a superset of W, however, it is not a proper superset of W because of Y= W.

Z is not a superset of W since it lacks the element “w” that is present in set W.

Differences Between Superset and Subset

The primary distinction between superset and subset is that they are diametrically opposed. Take, for example, two sets, M and N. M = {13, 15, 19} and N = {15, 19}. Then, {13, 15, 19} is the superset of {15, 19}. To put it another way, if M is the superset of N, then N is the subset of M. The distinctions between the superset and subset are listed below.

Properties of Superset

The following are the main qualities of a superset:

• Every set is a superset of itself.
• Each set is a subset of itself.
• A set has an endless number of supersets.
• Because the null set includes no items, we may claim that any set is a superset of an empty set, for example, every set H would be represented as H ⊃ φ
• Set B is the superset of set A if it is offered as a subset of set A.

Read More,

• Types of Sets
• Complement of Sets
• Representation of Sets
• Disjoint Sets

Solved Examples on Superset

Example 1: Assume M ={m, n, q, r, s} and N= {m, o, p, q}. Is M a subset of N or a superset? Provide reasons as well.

Solution:

Given: M ={m, n, q, r, s} and N= {m, o, p, q}.

Because the elements “o and p” in set N are not present in set M, the supplied set M is not a superset of set N. As a result, we can plainly state that M is not a superset of N. As a result, N does not constitute a subset of M.

Example 2: Determine who is a subset here if M = {x: x is an odd natural number} and N = {y: y is a natural number}.

Solution:

Given: M = {1, 3, 5, 7, 9, 11,13, …} and N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, …}.

It is obvious that set N contains ALL of the items of set M. As a result, set M is a subset of set N, or M ⊂ N.

Example 3: Show that M is the correct superset of N if M = {32, 33, 37, 39} and N= {32, 37, 39}. Justify your response.

Solution:

Given:

Set M = {32, 33, 37, 39}

Set N = {32, 37, 39}

M is the appropriate superset of set N since all of set N’s components are also present in set M, however, we can see that set M contains one more additional element (i.e., 33) than set N.

We can also show that set M is not equivalent to set N.

As a result, we may argue that set M is the correct superset of set N.

Example 4: There are three sets: a set of all real numbers (R), a set of natural numbers (N), and a set of all whole numbers (W). Determine the superset of them.

Solution:

Real numbers include natural numbers, whole numbers, integers, rational numbers, and irrational numbers, but not complex numbers.

Natural numbers include all positive counting numbers indicated by N = {1, 2, 3,…}

Whole numbers are made up of natural numbers plus 0. These are represented by the symbol W = {0, 1, 2, 3,…}

We can now conclude:

Real numbers ⊃ Natural numbers

Real numbers ⊃ Whole numbers

Also, Whole numbers ⊃ Natural numbers

Example 5: Check whether the following statements are true or false.

a) An empty set is a superset of every other set.

b) Every set is a superset of the empty set.

c) Every set is a superset of itself.

b) Every set has a limited number of supersets.

Solution:

a) False, as empty set do not contain any element.

b) True , as empty set is subset of all sets.

c) True, as set itself contains all the elements it have.

d) False, as we can add one element to set to make it superset, and that element can be anything.

Practise Problems on Supersets

Problem 1: Let A = {1, 2, 3, 4, 5} and B = {2, 4}. Determine whether each statement is true or false:

a) A is a superset of B.

b) B is a subset of A.

c) B is a proper subset of A.

d) A and B are disjoint sets.

Problem 2: Given the sets C = {red, green, blue} and D = {red, green, blue, yellow}. Is D superset of C or C superset of D?

Problem 3: Consider the sets E = {a, b, c, d}, F = {c, d, e}, and G = {a, e}. Draw a Venn diagram to represent these sets and their relationships. Identify any supersets or subsets.

Problem 4: Let H = {x, y}. Find the power set of H and identify which subset of H are supersets of the set {x}.

Problem 5: Let M be the set of all mammals and C be the set of all carnivores. Determine whether each statement is true or false:

a) M is a superset of C.

b) C is a subset of M.

c) M is a proper superset of C.

d) C is a proper subset of M.

FAQs on Superset

1. What is a Superset in Mathematics?

A set’s primary set is made up of the components of its subjects, which is known as a superset. In other words, a valid subset of A, such as set B, is a superset of A if B contains at least one element not found in A.

2. Is it Correct to Argue that Every Set is a Superset of an Empty Set?

Because the null set has no items, we may claim that every set is a superset of it.

3. What is the Superset Symbol?

The symbol “⊃” represents the connection between a superset and its subset. For example, the set O of odd numbers is a subset of the superset natural numbers N and may be represented as N ⊃ O.

4. What is the Distinction Between a Subset and a Superset?

Subsets and Supersets are essentially diametrically opposed. Set X can be considered to be a subset of Set Y if members of Set X are said to be contained in Set Y. If Set X contains {12, 13, 14, 15, 16, 17, 18} and Set Z contains {15, 16, 17}, we may say that Set Z is a subset of Set X and Set X is also a subset of Set Z.

5. What is a Proper Superset?

A Strict Superset is another name for a Proper Superset. Set X is considered to be a legitimate Superset of Y when it contains all of the elements of Set Y, and Set X must contain at least one element of Set Y.

6. How many Subsets are Possible for A Set?

There are two subsets of a set that have only one element. There are four subsets of a set that have two items. There are also eight subsets of a set having three items. For a set havind n elements, it has 2ⁿ number of subsets.