Superset

A superset is defined as a set of another smaller set if almost all elements of that smaller set are elements of the set. Set A is called the superset of another set B if all elements of set B are elements of set A. This superset relationship is denoted as A⊃B. For example if A is the set {a, b, c, d, e, f, g} and B is the set {a, b, c, d, e, f}, then A⊃B.

1.	What is a Superset?
2.	Properties of Superset
3.	Difference Between Superset and Subset
4.	Solved Examples on Superset
5.	Practice Questions on Superset
6.	FAQs on Superset

What is a Superset?

In set theory, Let A and B be any two sets. If B is contained in A or B ⊂ A, and A ≠ B, then A is a superset of B, represented as A⊃B. In other words, if a few or all the elements of set B are the elements of set A, then set A is considered as the superset of B. Let's consider the two sets to be, set A = {11, 12, 13, 14} and set B = {11, 13}. Since the elements of B {11, 13} are in set A {11, 12, 13, 14}, then we can say that set A is the superset of B but B is not a superset of A.

Superset: A is the superset of B

Superset Symbol

To represent the superset and its subset relationship, the symbol “⊃” is used. For instance, for the two sets X and Y, if X = {set of triangles} and Y = {set of obtuse triangles}. Then, X is the superset of Y because obtuse triangles are classified under the main category triangles. Thus, it can be denoted as X⊃Y and we can say that Y is a subset of X.

Properties of Superset

Given below are the major properties of a superset:

Since the null set contains no elements in it, thus we can say that every set is a superset of an empty set, such as for any set H, it would be represented as H ⊃ φ
If a set A is given as the subset of set B, then the set B should be the superset of the set A.

Difference Between Superset and Subset

The key difference between superset and subset is that the superset and subset are just opposite to each other. For example, if we take up two sets, M and N. M = {3, 5, 9} and N = {5, 9}. Then, {3, 5, 9} is the superset of {5, 9}. In other words, M is the superset of N, then N is the subset of M.

A superset is denoted using “⊃” whereas a subset is denoted using “⊂”. Thus, M ⊃ N and N ⊂ M.

Important Notes

A superset is a part of a universal set as it contains the elements of all the sets in the given context.
For a given superset, the set containing a few or most of its elements is called its subset.

Related Topics

Example 1: Determine the superset for the given set of numbers:
- Natural Numbers
- Whole Numbers
- Integers
- Rational Numbers
- Irrational Numbers
- Complex Numbers
Solution:

The set of real numbers R is the union of the set of rational numbers (Q) and the set of irrational numbers (Q'). Thus, we can say the set of real numbers, R = Q ∪ Q'. This further indicates that the set of real numbers include natural numbers, whole numbers, integers, rational numbers, and irrational numbers. Thus,
- R (Real Numbers) ⊃ N (Natural Numbers)
- R (Real Numbers) ⊃ W (Whole Numbers)
- R (Real Numbers) ⊃ Z (Integers)
- R (Real Numbers) ⊃ Q (Rational Numbers)
- R (Real Numbers) ⊃ Q' (Irrational Numbers)
Therefore, for natural numbers, whole numbers, integers, rational numbers, and irrational numbers, their superset is the set R, which is the set of real numbers.

On the other hand, complex numbers are the numbers that are neither rational nor irrational, and thus are not real numbers, thus the set of real numbers can't be considered as the superset for the set of complex numbers C. This implies, C ⊄ R. Therefore, for complex numbers, R is not their superset.
Example 2: Set A = {a, b, c, d, e, f, ...z} and set B = {a, e, i, o, u}. Identify the superset and the subset.

Solution:

Given, Set A = {a, b, c, d, e, f, ...z}, that is all the English alphabets and set B = {a, e, i, o, u}, that is all the five vowels that are a part of English alphabets only. Therefore, we can conclude that set A is the superset for subset B, that is A ⊃ B.

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FAQs on Superset

What is Superset in Maths?

A superset of a set is the main set consisting the elements of its subjects. In other words, a proper subset of A, say set B is a superset of a set A if all elements of A are in B but B contains at least one element that is not contained in A. For example, if A={5,15, 25} then B = {5, 10, 15, 20, 25} is a superset of A.

What is the Difference Between Universal Set and Superset?

A set A{a, b, c, d} is a superset of a set B {b, c, d} if all the elements of B are contained in A. In other words, every element of B is an element of A and can be written as B ⊃ A.

A universal set, denoted by 'U' contains the elements of all the sets in the given context. In other words, it is the superset of each of the given sets. For the three given sets, A{1, 3, 5}, B{2, 4, 6} and C{1, 3, 6}, the universal set, U = {1, 2, 3, 4, 5, 6}.

What is the Relation Between Superset and Subset?

A subset is a set that has either some or all of the elements of another set, called the superset. For C⊃D, we can say that C is a superset of D and D is the subset of C.

What is the Symbol of Superset?

The relationship between a superset and its subset is represented by the symbol “⊃”. For example, the set O of odd numbers is a subset for the superset numbers N and can be represented as N⊃O.

What is a Strict Superset?

A strict superset refers to a superset that has at least one element which does not exist in its subset. For example, A{♢,♡,♣,♠} is the strict superset of B{♢,♣,♠}.

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