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.
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=16Hence, 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.
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)
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