In this post we will learn about the concept of subset with properties and examples.
To fully understand the chapter, you should have basic knowledge about sets and its properties.
What is a Subset in math?
Suppose two sets A & B are given.
If all the elements of set B is present in set A, then set B is called subset A.
If all elements of set A is present in set B, then set A is subset of B.
Let’s try to understand the concept in simple manner.
Consider set A as a master set containing 5 elements.
A = { 3, 6, 9, 10, 12}
Now from master set A, we have copied some of its elements to form set B.
B = { 6, 10, 12 }
Since all elements of B are taken/copied from A, the set B is a subset of A.
Representing Subset
Subset is represented by symbol \mathtt{\subseteq }
So, if set B is subset of set A then we can write; B \mathtt{\subseteq } A
The expression B \mathtt{\subseteq } is read as ” B is subset of A ” or ” B is contained in A”
Properties of Subset
(01) Every set is a subset of itself.
Explanation:
Consider set A with following elements;
A = { 9, 7, 13, 10 }
Making a copy of set A, we get;
A’= A = { 9, 7, 13, 10 }
See that set A’ is subset of set A since all elements of A’ is present in set A.
Hence set A is subset of set A (i.e.) \mathtt{A\ \subseteq \ A} .
(02) Empty set is a subset of every set.
Explanation
Consider set A with following elements;
A = { 3, 6, 5 }
Now consider empty set B;
B = { }
Since B is an empty set with no elements, it is subset of set A or any set whatsoever.
(03) For a set containing n elements, the total number of subset possible is given by formula \mathtt{2^{n} \ }
Explanation
Consider set A with 2 elements;
A = { 14, 17 }
The number of possible subsets are:
\mathtt{\Longrightarrow \ 2^{n} \ }
Here number of elements n = 2
\mathtt{\Longrightarrow \ 2^{2}}\\\ \\ \mathtt{\Longrightarrow \ 4\ \ }
Hence, there can be 4 possible subsets for set A.
Listing all the subsets of A
(i) Empty set P = { }
(ii) Q = {14}
(iii} R = {17}
(iv) S = {14, 17}
Types of Subset
There are two types of subsets;
(a) Proper Subset
(b) Improper Subset
Proper Subset
Proper subset B contains elements of the main set A, but it should not contain all the elements so that B becomes equal to A.
Given below are characteristics of Proper Subsets
⟹ subsets containing all the elements of the main set are not considered as proper subset
⟹ Proper subset contain at least one element and not all the elements of the main set.
⟹ The number of proper set can be found using the formula \mathtt{2^{n} -1}
⟹ Proper Subset is denoted by symbol ” \mathtt{\subset } “
Examples of Proper Subset
Example 01
Given below is the set A with three elements.
A = { 1, 3 , 5 }
Find all the proper subsets for A
Solution
Given below are the proper subsets for given set A.
⟹ { }
⟹ { 1 }
⟹ { 3 }
⟹ { 5 }
⟹ { 1, 3 }
⟹ { 3, 5 }
⟹ { 1, 5 }
hence, the above 7 sets are the proper subset of A.
Example 02
Consider the set A with four elements
A = { 10, 22, 35, 41 }
Check which of the below set B, C & D is a proper subset
B = { 35, 22, 10, 41 }
C = { 10, 41 }
D = { 6, 22 }
Solution
Set B is not the proper subset because it contains all the elements of set A.
Set C is the proper subset as it contains two elements of set A
D is not even a subset of A as it contains element ” 6 ” which is not present in set A.
Improper Subset
A subset that contains all the elements of the master set is called improper subset.
⟹ Improper subset is the exact replica of the given master set.
⟹ There is only one possible improper subset of a given set.
Examples of Improper Subset
Example 01
Consider the set A with 5 elements
A = { 9, 22, 73, 45, 31 }
Check which of the following below set is Improper subset
B = { 73, 22, 9 }
C = {31, 73, 22, 9, 45 }
Solution
Set B contains three element of set A.
Hence set B is a proper subset of A ( i.e. B ⊂ A )
Set C contains all the element of set A.
Hence it is improper subset of A.
Subsets -Solved Problems
(01) Find the number of possible subsets for below set A
A = {13, 29, 37 }
Also list all the possible subsets for the set A.
Solution
The number of element in set A is 3.
Formula for number of subset is \mathtt{2^{n}}
Here n = 3.
Putting the value of n in the formula.
\mathtt{\Longrightarrow \ 2^{3}}\\\ \\ \mathtt{\Longrightarrow \ 8}
Hence, there are total of 8 subsets possible for the given set.
Listing all the 8 subsets below:
⟹ { }
⟹ { 13 }
⟹ { 29 }
⟹ { 37 }
⟹ {13, 29 }
⟹ { 29, 37 }
⟹ { 13, 37 }
⟹ { 13, 29, 37 }
(02) Given below is the set A with 6 elements.
A = { 16, 37, 29, 82, 101, 114}
Check if the below sets are subset of A or not?
(i) B = { 37, 42 }
(ii) C = { 16, 82, 114 }
(iii) D = { }
(iv) E = {114, 29, 15}
(v) F = { 101, 37 }
Solution
(i) B is not subset of A as it contains element 42 which is not present in set A.
(iii) C is subset of A (C ⊆ A) as it contains all the element present in set A.
(iii) D is an empty set.
We know that empty set is a subset of any possible set.
Hence, D ⊆ A.
(iv) Set E is not a subset of A since it contain additional element 15 which is not present in set A.
(v) F is subset of A since both elements, 101 & 37 is present in set A.
Hence, F ⊆ A
(03) Given below is the set A with 5 elements
A = { 33, 21, 63, -10, 5 }
Check if the below sets are proper or improper subsets?
(i) B = { 5 , 33 }
(ii) C = { -10, 21, 33, 31 }
(iii) D = { 63, 21, 33, 5 }
(iv) E = { 5, 21, 63, -10, 33}
Solution
(i) Set B is a proper subset of A as it contain elements 5 & 33 which is present in set A.
(ii) Set C is not even a subset of A.
It contains element 31 which is not present in set A.
(iii) Set D is a proper subset of A.
All the element in set D is present in set A.
(iv) Set E is a improper subset of A.
Set contains all the elements of A. Since no element is left between set A & E, the set E is called improper subset.