Mathematics often uses sets to organize objects, numbers, or even ideas. A set is simply a well-defined collection of objects, such as numbers, people, or even books. Sets can be of many types — finite sets, infinite sets, universal sets, power sets, and more. Among these concepts, the superset plays a very important role in understanding relationships between sets.
In this article, we will learn what a superset in maths is, how to identify one, its properties, symbols, and solved examples to make the concept crystal clear.
Superset Meaning
In set theory, if every element of Set X belongs to Set Y, then Y is called the superset of X. This means that X is entirely contained in Y, but the two sets are not equal. In such a case:
- X is a proper subset of Y (written as X ⊂ Y)
- Y is the superset of X (written as Y ⊃ X)
In other words, the number of elements in X is always less than the number of elements in Y.
Superset Example
Let’s look at an example:
- Set Y = {1, 2, 3, 4, 5}
- Set X = {1, 3, 5}
Since every element of X is also present in Y, we can say that Y is a superset of X.
Here,n(X) = 3 and n(Y) = 5, so n(X) < n(Y).
Superset Definition
Formally, a superset is defined as:
Set A is a superset of Set B if all the elements of B are also present in A.
Example:
- Set A = {1, 2, 3, 4}
- Set B = {1, 3, 4}
Here, A ⊃ B because B’s elements are already in A. However, B is not a superset of A.
What is a Superset in Math?
A superset in maths can be thought of as a larger container of elements. If Set A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and Set B = {1, 3, 5, 7, 9}, then all elements of B appear in A. Therefore, A is a superset of B.
Superset Symbol
The superset is denoted using the symbol ⊃.
If A is a superset of B, we write A ⊃ B.
Example:
- X = {All polygons}
- Y = {Irregular polygons}
Here, X ⊃ Y since all irregular polygons belong to the larger set of polygons. Similarly, we can say Y ⊂ X (Y is a subset of X).
Proper Superset (Strict Superset)
A proper superset (or strict superset) contains all elements of another set plus at least one additional element.
Example:
- A = {a, b, c}
- B = {a, b, c, d}
Here, B is a proper superset of A because it has all the elements of A and one extra element (d).
Key points:
- If Set C = {a, b, c}, then C is a superset of A, but not a proper superset, since C = A.
- If Set D = {a, b, e}, then D is not a superset of A, since it misses “c”.
Properties of Supersets
Some important properties include:
- Every set is a superset of the empty set (φ), since φ has no elements.
- Example: A ⊃ φ.
- Example: A ⊃ φ.
- If A ⊂ B, then B ⊃ A.
- The null set (φ) has only one subset — itself. Thus, it cannot be a proper superset of any set.
- A superset is always part of the universal set in the given context.
Superset vs Subset
The difference between supersets and subsets lies in perspective:
- Superset (⊃): The larger set that contains another set.
- Subset (⊂): The smaller set that is part of the larger set.
Example:
- M = {4, 7, 9}
- N = {4, 9}
Here, M ⊃ N (M is the superset of N), while N ⊂ M (N is the subset of M).
Solved Examples on Supersets
Example 1:
Let A = {a, e, i, o, u} and B = {a, c, d, e}.
- Since B has elements (c, d) not present in A, A is not a superset of B.
Example 2:
A = {2, 3, 7, 9}, B = {2, 7, 9}
- All elements of B are in A, and A has one extra element (3).
- Therefore, A is a proper superset of B.
Conclusion
Supersets are an essential part of set theory in mathematics, helping us understand how different sets relate to each other. A superset simply means one set contains all the elements of another set, with or without extra elements. By learning about supersets, proper supersets, and their properties, students can easily distinguish them from subsets. This concept not only strengthens logical thinking but also lays the foundation for advanced topics in algebra, probability, and computer science.
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FAQs on Supersets
Ans: The superset is denoted by ⊃.
Ans: Yes, a set is always an improper superset of itself.
Ans: A superset that has at least one additional element compared to the other set.
Ans:Yes, because the empty set has no elements to exclude.
Ans: If A ⊃ B, then B ⊂ A. Supersets contain, subsets are contained.
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