## What Is a Superset?

We call set B a superset of set A if B contains all the elements of A. In other words, if A is a subset of B, then B is the superset of A.

We know that set A is called a subset of set B if all the elements of A are present in B. Then what does superset mean? Let’s understand this with a Venn diagram.

In the following figure, set B contains set A.

A is a subset of B.

B is a superset of A.

## Superset Meaning

In mathematics, a superset is a set that consists entirely of the elements of the smaller set. If P is the superset of Q, we can infer that Q is the smaller set and P contains all elements of Q.

## Superset Definition

**A superset of a given set is a set that contains all the elements of the given set.**

If B is a superset of A, then A is a subset of B.

**Example: **A = {1, 2}, B= {1, 2, 3, 4, 5}

B contains all the elements of A.

Thus, B is a superset of A.

Also, A is a subset of B.

## What Is a Proper Superset?

The **proper superset** or a **strict superset** is a superset that contains all the elements of the smaller set but also has some extra elements of its own.

If 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 that is not present in set A.

Suppose we have two sets:

A={1, 3, 5}

B={1, 3, 4, 5}

Here, B is a superset of A.

We can see that B is not exactly equal to A.

B contains the element 4, which is not present in A.

Such a superset is called a proper superset.

## Superset Symbol

The symbols “⊃” or “⊇” are used to denote the superset.

- B ⊃ A means that B is a proper superset or a strict superset of set A such that A B (the sets are not equal.)

- B ⊇ A means that B is a superset of A, but the possibility of A = B is also considered.

## Properties of Superset

- Empty set contains no elements. So, every set is a superset of the empty set.

For any set A, we have A ⊃ ∅.

- Every set is a superset of itself.

For any set A, we have A ⊇ A.

- For the given two sets (P and Q), if P ⊂ Q, then P ⊃ Q is true, indicating that subset and superset are opposite to one another.
- If B is a superset of A, then

A ∩ B = A

A ∪ B = B

- The cardinality of the superset is always greater than or equal to the other set.

## Difference between Superset and Subset

Subset | Superset |
---|---|

If A is a subset of B, the set A is contained in B. | If B is a superset of A, then B contains A. |

A ⊂ B means that A is a proper subset of A. | B ⊃ A means that B is a proper or strict superset of A. |

Subset is a smaller set compared to the other set. | Superset is a bigger set compared to the other set. |

If A ⊂ B, then B ⊃ A. | If B ⊃ A, then A ⊂ B. |

The empty set is a subset of every set. | Every set is a superset of the empty set. |

Every set is a subset of itself. | Every set is a superset of itself. |

## Facts on Superset

- A superset contains all the elements of the smaller set (it may or may not contain some extra elements as well.)
- The set of real numbers is a superset of the set of integers, the set of natural numbers, the set of rational numbers, the set of whole numbers.
- The set of whole numbers is a superset of the set of natural numbers.

N = {1, 2, 3, 4, …}

W = {0, 1, 2, 3, 4, …} - If A ⊂ B and B ⊃ A, then A = B

## Conclusion

A superset is a set that contains almost all of the components of a smaller set that it is composed of. We understand that if B is contained within A, then A includes B. The following solved examples and practice problems will further ease understanding of the concept of supersets.

## Solved Examples on Superset

**1**.** Let ****Y= {1, 2, 3, 4, 5}**** and ****X= {1, 3, 5}****. Identify the superset.**

**Solution:**

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

Every element of X is also an element of Y.

The set Y contains the set X.

Y ⊃ X

Hence, Y is the proper superset of X.

**2**. **E is the set of positive even integers. Write two supersets of E.**

**Solution:**

E is the set of positive even integers.

E = {2, 4, 6, 8, 10, …}

We know that the set of integers is given by

Z = {…, -3, -2, -1,0, 1, 2, 3, 4, …}

Thus, Z is a superset of E.

The set of natural numbers is given by

N = {1, 2, 3, 4, 5, …}

Thus, N is a superset of E.

**3**.** Let P is a set of all polygons. Identify the sets for which P will be a superset. **

**i) Set of convex polygons**

**ii) Set of quadrilaterals**

**iv) Set of triangles **

**v) Set of circles**

**vi) Set of 2D shapes**

**Solution: **

P is a set of all polygons.

In geometry, a polygon can be defined as a flat or plane, two-dimensional closed shape bounded with straight sides. It does not have curved sides. Also, we need at least three sides to form a polygon.

i) Set of convex polygons will be a subset of the set P. Thus, P is a superset of the set of convex polygons.

ii) All quadrilaterals are polygons. So, the set P is a superset of the set of quadrilaterals.

iii) Triangles are the polygons with three sides. So, P is a superset of the set of triangles.

iv) Circles are not polygons since they are not formed by straight lines. Thus, P is not a superset of the set of circles.

v) Polygons are 2D shapes, but 2D shapes contain polygons and many other shapes like circles. So, the set of polygons is a subset of the set of 2D shapes. The set of 2D shapes is a superset of the set of polygons.

**4**.** If ****A = {4, 5, 7, 8}**** and ****B = {4, 7, 8}****, justify why A is a proper superset of B. **

**Solution:**

A = {4, 5, 7, 8}

B = {4, 7, 8}

A is the proper superset of B because all elements of B are in A. The set A also has one extra element, 5, which is absent in B.

## Practice Problems on Superset

## Superset in Maths: Definition, Symbol, Properties, Example

### If set C is a proper superset of B, then ______.

If C is a proper superset of B, then we represent it as $C \supset B$.

### If X = {1, 2, 3}, Y= {}, Z = {1, 2}, then which of the following is true?

Every set is a superset of the empty set.

Also, here Z is a subset of X. Thus, X is a superset of Z.

### Which of the following statements is not true?

The set of real numbers is the superset of the set of rational numbers.

## Frequently Asked Questions about Superset

**What is a proper subset?**

If A is a proper subset of B, then all the elements of A are present in B, but A is not equal to B (B contains at least one extra element).

**What are the symbols for a superset and a subset?**

The superset is denoted by the sign “⊃” and the subset by the symbol “⊂.”

**What is the universal set?**

Universal set is a set that contains all the elements of all possible subsets under discussion.

**Is every set a superset of an empty set?**

Yes, since a null set has no elements, we can say that every set is thought of as the superset of a null set.

**Can a set be a superset of itself?**

Yes, a set is always thought of as a superset of itself since every set has all of its own elements.