Superset
In mathematics, a superset is a set that contains all the elements of another set, which is called the subset. We know that if B lies inside A, then it means that A contains B. In other words, if B is a subset of A, then A is the superset of B.
More precisely, if set B is a subset of set A, then A is a superset of B. Let us learn more about supersets along with the difference between supersets and subsets.
| 1. | What is a Superset? |
| 2. | Superset Symbol |
| 3. | Properties of Superset |
| 4. | Differences Between Superset and Subset |
| 5. | Solved Examples on Superset |
| 6. | Practice Questions on Superset |
| 7. | FAQs on Superset |
What is a Superset?
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}, then A is a superset of B which is written as A⊃B. Note that, here the set B is a subset of the set A.
Let A and B be any two sets. If B is contained in A or "B is a subset of A (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 (whereas B is a subset of A).
Here are more examples of supersets in maths:
- Set of real numbers is a superset of each of set of rational numbers, set of irrational numbers, set of integers, set of natural numbers, set of whole numbers etc.
- Set of integers is a superset of set of even integers.
- Set of natural numbers is a superset of set of prime numbers.
- Set of square matrices is a superset of set of invertible matrices.
Superset Symbol
To represent the superset and its subset relationship, the symbol “⊃” is used. In fact, we have two superset symbols.
- ⊇, which is read as "superset or equal to" (or) sometimes simply as "superset of".
- ⊃, which is read as "superset of". This means it represents "strictly superset of" but NOT "equal to".
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 types of triangles. Thus, it can be denoted as either X⊇Y or X⊃Y. But there is a minor difference between these two symbols and its usage depends on the type of superset. There are two types of supersets:
- Proper superset is represented by '⊃' (where X⊃Y but X is strictly NOT equal to Y)
- Improper superset is represented by '⊇' (Where X may or may not be equal to Y)
Properties of Superset
Given below are the major properties of a superset:
- Every set is a superset of the empty set.
- Every set is a superset of itself.
- The number of supersets of a set is infinite.
- 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 set B is the superset of set A.
Differences 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. The differences between the superset and subset are tabulated below.
| Superset | Subset |
|---|---|
| If set A contains set B, then A is said to be the superset of B. | If set A is contained in set B then A is said to be the subset of B. |
| It is denoted by the symbol A⊃B or A⊇B. | It is denoted by the symbol A⊂B or A⊆B. |
| If A⊃B then B⊂A. | If A⊂B then B⊃A. |
| Every set is a superset of the empty set. | The empty set is a subset of every set. |
Important Notes on Superset:
- A superset of a set contains all the elements of the set (it may contain some more elements extra also).
- For a given superset, the set containing a few or most of its elements is called its subset.
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FAQs on Superset
What is the Meaning of Superset in Math?
A superset of a set is the set consisting all elements of the given set. For example, if A = {5,15, 25} then B = {5, 10, 15, 20, 25} is a superset of A.
What is the Definition of a Strict Superset?
A strict superset (also known as a proper 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{♢,♣,♠}.
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 A ⊃ B (A is superset of B).
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} in a given context, the universal set, U = {1, 2, 3, 4, 5, 6}.
Why are Subsets and Supersets Important in Maths?
Supersets and subsets are important concepts in mathematics because they help to establish relationships between different sets of objects, and can be used to define and prove various mathematical properties and theorems.
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 natural numbers N and can be represented as N⊃O.
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 in this case, D is the subset of C.
Can a Set be a Superset of Itself?
Yes, a set is considered as a superset of itself always, since every set contains all its own elements.