Superset in Set Theory Explained with Symbol and Meaning

A set is a well-defined collection of any mathematical objects in mathematics. Anything can be included in the collection of objects. It could be a collection of people, a group of statistics, or something else. Finite sets, infinite sets, power sets, universal sets, and so on are examples of different types of sets. In this article, we are going to discuss what is a superset with superset meaning along with many solved examples.

Superset Meaning

If X ⊆ Y and X ≠ Y then set X is said to be a proper subset of set Y.

We write X ⊂ Y in symbols.

Y is referred to as the superset of X in this case.

To put it another way, every element of X is also an element of Y, despite the fact that X and Y are not equal. In other words, the number of X elements is smaller than the number of Y elements.

Superset Example

Let Y  =  {1, 2, 3, 4, 5} and X  =  {1, 3, 5}

Every element of X is also an element of Y in the two sets above, and the number of elements of X is less than the number of elements of Y.

That is, n(x)  =  3 and n(Y)  =  5 -----> n(x) < n(Y) 

Hence, Y is the superset of X.

Superset Definition

Let’s understand Superset Definition

Set A is considered the superset of B in set theory if all of the components of set B are also elements of set A. For example, if set A contains {1, 2, 3, 4} and set B contains {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 also present in set A. So, we can also say that B is not a superset of A.

What is a Superset in Math?

A superset in math can be defined as a set of elements containing all of the elements of another set. In other words, A is a superset of B if it contains all of the elements of B.

For example, consider the following two sets:

Set A elements are  {1,2,3,4,5,6,7,8,9,10}

Set B elements are {1,3,5,7,,9}

If we compare set A and set B, we can notice that all of the elements of B are in the set A. As a result, we say that A is a superset of the set B.

Superset Symbol

The superset is represented by using the symbol “⊃”. For example, the set A is the superset of set B, then it is symbolically it is represented as A ⊃ B.

Let’s 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 Y and represented as (Y⊂X).

Proper Superset

The proper superset is also called a strict superset. If set B is the correct superset of set A, then it contains all of the items of set A, but it must also have at least one member not found 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}

We can conclude from the above given sets,

B is the proper superset of A, as all elements of 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, because set C= set A

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

Properties of Superset

Following are the important properties of superset:

Every set is a superset of an empty set, and we know that the null set contains no elements.

Therefore, if A is the set, then it is represented as A ⊃ φ

Also, if A is the subset of B i.e (A ⊂ B), then B should be the superset of set A i.e (B⊃A).

If null set is a superset

If the null set is a superset, then it has only one subset. That is { }. 

More clearly, the null set is the only subset to itself and also it is not a proper subset.

Because, { }  =  { }

As a result, a null set is a set that has only one subset.

Important Notes

A superset is a subset of a universal set because it comprises members from all of the sets in the context.

The set containing a few or most of the members of a particular superset is referred to as its subset.

Difference Between Superset and Subset

The main difference between a superset and a subset is that they are in opposition to one another. Let's consider that M and N are the two sets. We can state that N is the subset of M if M is the superset of N.

Let the two set be M = {4, 7, 9} and N = {4, 9}

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

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

The symbol used to represent the superset is “⊃”, whereas “⊂” is  used to denotes the subset

Solved Examples on Superset

Question 1. Let A ={a, e, i, o, u} and B= {a, c,d, e}. Is A a subset or a superset of B? Also, provide justification.

Solution: Given:

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

B = {a, c, d, e}

Here, the given set A is not a superset of set B because the elements “c and d”which is in set B that are not present in set A. Hence, clearly we can say A is not a superset of B. As a result, B is not a subset of A.

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

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 also present in set A, but we can see that set A has one more extra element (i.e., 3) when compared with set B.

Also, we can see set A is not equal to set B.

Hence, we can say set A is the proper superset of set B.