Subset and Superset (Definition, Examples, Symbol, Difference)

Subset

What is Subset of a Set?

Set A is called a Subset of set B if all the elements of set A are present in set B. Also, the set B, in this case, can be called the Superset of the set A.

Subset Symbol

If set A is a Subset of set B, then mathematically, it can be written as,

A\subseteq B

Example of Subset

To understand the concept of Subset of a Set, let us consider two sets A and B as given below,

A = \{1, 2, 3\}

B = \{1, 2, 3, 4, 5, 6\}

From the above two sets A and B, it can be observed that all the elements of set A are also a part of set B. In other words, set B contains all the elements of A as well as some other elements. Therefore, set A, in this case, can be called the Subset of set B and B can be called Superset of the set A.

The figure below represents the Venn Diagram for A Subset of B for the above example.

Subset

From the above figure, it is clear that all the elements of set A are also the elements of set B. Therefore, set A can be called a Subset of set B. Mathematically it can be written as,

A \subseteq B

Types of Subset

The Subsets can be divided into two types as follows:

  • Proper Subsets

For the above example, set B has all the elements of set A as well as some other elements in it. Therefore, set A can be called a Proper Subset of Set B.

  • Improper subsets

If two sets, say A and B, are Equal Sets, that means both the sets have the same number of elements as well as the same elements, then set A is also a Subset of the set B. The Subset, in this case, can be called Improper Subset of the set B or vice versa.

Properties of Subset

If set A is a Subset of Set B, then ,

  • The Intersection of set A with set B is set A itself. Mathematically,

A \cap B = A

  • The Union of set A with set B is set B.

A \cup B = B

  • The Cardinality of Intersection of set A with set B is equal to the cardinality of set A

n(A\cap B)=n(A)

Maximum Number of Subset of a Set

If a set has n number of elements, then the maximum number of Subsets that the Set can have is 2^n.

For Example, Let us consider a set A=\{a, b, c\}. The set A has 3 elements. Therefore, the maximum number of Subsets of A is 2^3=8 and the Subsets of A are,

\{\},\,\{a\},\,\{b\},\, \{c\},\, \{a,\, b\},\, \{b,\, c\},\, \{c,\, a\},\, \{a,\, b,\, c\}

How Do you Find The Subset of a Set

Let us consider a set B=\{p, q, r, s\}

Before finding the Subset of a Set we need to find out how many Subsets that the given set can have. The maximum number of Subsets that a set can have is 2^n, where, n is the number of elements of the given set.

Here, n=4

\therefore 2^n=16

Hence, the given set B can have a maximum of 16 numbers of Subsets.

We know that, the Empty set \phi written as \{\} is a Subset of every set.

Therefore, the subset of the given set will be,

Sets having a no element or Single element: \{\},\, \{p\},\, \{q\},\, \{r\},\, \{s\}

Sets having two elements: \{p, q\},\, \{p, r\},\, \{p, s\},\, \{q, r\},\, \{q, s\},\, \{s, r\}

Sets having three elements: \{p, q, r\},\, \{p, r, s\},\, \{p, q, s\},\, \{q, r, s\}

Sets having four elements: \{p, q, r, s\}

Superset

Superset Definition

A set B is called a Superset of set A if all the elements of set A are present in set B. Therefore, the Superset is the opposite of a Subset.

Symbol of Superset

If set B is a superset of set A, then mathematically, it can be written as,

B\supseteq A

Example of Superset

To understand the concept of Superset of a Set, let us consider the same sets A and B as given below.

A = \{1, 2, 3\}

B = \{1, 2, 3, 4, 5, 6\}

The set B has all the elements of set A as well as some extra elements. Therefore, set B is a Superset of set A. The figure below shows the Venn Diagram of Superset of a Set.

Superset

As shown in the above figure, all the elements of set A are present in set B. Therefore, B is a Superset of set A.

Mathematically,

B\supseteq A

Frequently Asked Questions (FAQ)

Set A is called a subset of set B if all the elements of set A are present in set B.

A set B is called a superset of set A if all the elements of set A are present in set B.

The symbol of Subset and Superset is similar, however, the method of writing is different. For example, if a set A is a subset of set B then it can be written as,

A ⊆ B

For the above case, it can also be said that B is a superset of set A. Then, it can be written as,

 B ⊇ A

A ⊆ B can be read as A is a subset of B.

Yes, an empty set is a subset of every set.

If n is the number of elements of a set A, then there will be 2^n number of subsets of the set A.

If a set has 2 elements, then the possible number of subsets of the set will be 2^2=4

Let,
A={a, b, c, d, e}
B={a, c, e}
Here, B is called the subset of set A as all the elements of B are also the elements of A. A is called the superset of set B.

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