Superset

In Mathematics, a set is defined as a well-defined collection of any mathematical objects. The collection of objects can be anything. It can be a group of people, a group of numbers and so on. There are different types of sets, such as finite sets, infinite sets, power sets, universal sets, etc. In this article, we are going to discuss one of the types of sets called “Superset” with a complete explanation along with many solved examples.

Table of Contents:

Superset Definition

In set theory, set A is considered as the superset of B, if all the elements of set B are the elements of set A. For example, if set A = {1, 2, 3, 4} and set B = {1, 3, 4}, we can say that set A is the superset of B. As the elements of B [(i.e.,)1, 3, 4] are in set A. We can also say that B is not a superset of A.

Superset Symbol

The superset relationship is represented using the symbol “⊃”. For instance, the set A is the superset of set B, and it is symbolically represented by A ⊃ B.

Consider another example,

X = {set of polygons}

Y = {set of irregular polygons}

Then X is the superset of Y (X⊃Y). In other words, we can say that Y is a subset of X (Y⊂X).

Proper Superset

The proper superset is also known as a strict superset. The set B is the proper superset of set A, then all the elements of set A are in B, but set B must contain at least one element which is not present in set A.

For example, let us take four sets.

A = {a, b, c}

B = {a, b, c, d}

C= (a, b, c}

D = {a, b, e}

From the sets given above,

B is the proper superset of A, as B is not equal to A

C is a superset of set A, but the set C is not a proper superset of set A, as C= A

D is not a superset of A, as the set D does not contain the element “c” which is present in set A.

Properties of Superset

The important properties of superset are:

  • Every set is considered as a superset of an empty set, as the null set has no elements in it. If A is the set, then it is represented as A ⊃ φ
  • If A is the subset of B (A ⊂ B), then B should be the superset of set A (B⊃A).

Difference Between Superset and Subset

The major difference between superset and subset is that the superset and subset are opposite to each other. Let us assume, M and N are the two sets. If M is the superset of N, then we can say that N is the subset of M.

(i.e) M = {4, 7, 9} adn N = {4, 9}

Thus, {4, 7, 9} is the superset of {4, 9}.

In other words, {4, 9} is the subset of {4, 7, 9}

The symbol to represent the superset is “⊃”, and the symbol that denotes the subset is “⊂”.

Also, read: Types of sets.

Problems on Superset

Go through the following problems to understand the concept of the superset.

Example 1:

Let A = {a, e, i, o, u} and B = {a, c,d, e}. Is A a superset of B? And also, give reason.

Solution:

Given:

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

B = {a, c, d, e}

The set A is not a superset of set B because the elements “c and d” in set B are not present in set A. Hence, A is not a superset of B. We can also say that B is not a subset of set A.

Example 2:

If A = {2, 3, 7, 9} and B= {2, 7, 9}, then A is the proper superset of B. Justify.

Solution:

Given:

Set A = {2, 3, 7, 9}

Set B = {2, 7, 9}

Here A is the proper superset of set B, as all the elements of set B are in set A, but set A has one more extra element (i.e., 3) than set B.

Also, set A is not equal to set B.

Hence, A is the proper superset of set B.

Practice Problems

Solve the following problems on superset:

  1. Is {7, 9, 8, 12} a superset of {7, 12}. Justify your answer.
  2. Is {4, 11, 12} a proper superset of {4,11, 12}. Also, justify your answer.

Frequently Asked Questions on Superset

What is meant by superset?

Set A is said to be the superset of B if all the elements of set B are the elements of set A.

Define Proper superset.

Set A is said to be the proper superset of set B, then all the elements of set B are in A, but A should contain at least one element that is not present in set B.

Mention the symbols to represent the superset and subset.

The symbol to represent the superset is “⊃” and the subset is “⊂”.

Is {2, 3, 9} a superset of {2, 4}?

No, {2, 3, 9} is not a superset of {2, 4}, as the number “9” is not present in the set {2, 4}.

Can we say every set is a superset of an empty set?

Yes, we can say every set is considered as the superset of a null set, as the null set has no elements in it.

To learn more about different sets, register with BYJU’S – The Learning App and watch many interesting videos to quickly understand the concept.

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