Subsets of a Given Set

Subsets of a Given Set – Definition, Symbol, Properties | How to find the Number of Subsets in a Set?

Subsets of a Given Set mean the subset having the elements present in a set. In other words, Subsets are a part of the set. A set is nothing but a collection of elements placed within curly braces. The example of a set is {a, b, c}. If we take a set of even numbers and name it as A and set B has {2, 4, 6}, then B is the subset of set A. It is represented as B⊆A. The elements of sets may be variables, groups of real numbers, constants, or whole numbers, etc.

What is Subset of a Set?

The subset is said to be a part of the set. Subset must have the elements that present in a set. If set A is said to be a subset of Set B, then all the elements of Set A are must present in Set B. Set A is contained inside Set B.
Example: If Set A has {C, D} and set B has {C, D, E}, then A is the subset of B because all the elements of A are also present in set B. The subset is denoted by the symbol ⊆ we read as ‘subset of’. A ⊆ B; which means Set A is a subset of Set B.

All Subsets of a Set

The subset of any given set must consist of all possible sets along with the elements and also having a null set. Check out the below example to understand the subset of a set.
Example: A = {7, 6, 8, 9}
Solution: Given that A = {6, 7, 8, 9}.
Subsets = {}, {6}, {7}, {8}, {9},
{6,7}, {6,8}, {6,9}, {7,8},{7,9}, {8,9},
{6,7,8}, {7,8,9}, {6,8,9}, {6,7,9}
{6, 7, 8, 9}

Types of Subsets

Mainly, there are two different types of subsets available. they are classified as

  • Proper Subset
  • Improper Subsets

Proper Subset

The proper subset contains some elements of an original set along with a null set. Set A is treated to be a proper subset of Set B if Set B has at least one element that is not present in Set A. A proper subset is denoted by ⊂ and it is read as ‘is a proper subset of’. We can show a proper subset for set A and set B as A ⊂ B.

Proper Subset Formula

If we take n number of elements from a set having N number of elements, then it shows as  NCn number of ways. The number of possible subsets having n number of elements from a set containing N number of elements is equal to  NCn.

Also, Read:

Improper Subset

A subset that has all elements of the original set is called an improper subset. It is denoted by ⊆. The Improper Subset consists of all elements of a set that doesn’t miss any element.

Example: Set P = {3, 5, 7}
The subsets of P are {}, {3}, {5}, {7}, {3,5}, {5,7}, {3,7} and {3,5,7}.
Where, {}, {3}, {5}, {7}, {3,5}, {5,7}, {3,7} are the proper subsets and {3,5,7} is the improper subsets. Therefore, we can write {3,5,7} ⊆ P.

Note: The empty set is an improper subset of itself but it is a proper subset of any other set.

Power Set

The power set is defined as the collection of all the subsets. It is represented by P(A).
If A is set having elements {a, b}. Then the power set of A will be
P(A) =  {∅, {a}, {b}, {a, b}}

How many Subsets and Proper Subsets does a Set have?

Let us consider a set that consists of n elements, then the number of a subset of the given set is 2n and the number of proper subsets of the given subset is given by 2n-1.

Example: If set A consists the elements, A = {c, d}, then the proper subset of the given subset are {}, {c}, {d}.
The number of elements in the set is 2.
The formula to find the number of proper subsets is 2n – 1.
= 2² – 1
= 4 – 1
= 3
Thus, the number of proper subset for the given set is 3 ({ }, {c}, {d}).

Properties of Subsets

The below are some of the important properties of subsets.

  • Every set is considered as a subset of the given set itself. It means that X ⊂ X or Y ⊂ Y, etc
  • Also, we can say, an empty set is considered as a subset of every set.
  • A is a subset of B. It means that A is contained in B
  • If a set A is a subset of set B, we can say that B is a superset of A

Solved Examples on Subsets of a Given Set

1. If A {2, 3, 7}, then write all the possible subsets of A. Find their numbers?

Solution:
Given that set A {2, 3, 7}.
Let us write all the possible subsets of a given set.
The subset of A containing no elements – {  }
The subset of A containing one element each – {2} {3} {7}
The subset of A containing two elements each – {2, 3} {2, 7} {3, 7}
The subset of A containing three elements – {2, 3, 7}
Therefore, all possible subsets of A are { }, {2}, {3}, {7}, {2, 3}, {2, 7}, {3, 7}, {2, 3, 7}
Therefore, the number of all possible subsets of A is 8 which is equal to 2³.
Proper subsets are = {  }, {2}, {3}, {7}, {2, 3}, {3, 7}

Number of proper subsets are 7 = 8 – 1 = 2³ – 1.

2. If the number of elements in a set is 4, find the number of subsets and proper subsets.

Solution:
Number of elements in a set = 4
Then, number of subsets = 24 = 16
Also, the number of proper subsets = 24 – 1
= 16 – 1 = 15

3. If A = {5, 6, 7, 8, 9}

The formula to calculate the number of proper subsets of a given set is 2n – 1
then the number of proper subsets = 25 – 1
= 32 – 1 = 31   {Take [ 2n – 1]}
and the formula to calculate the number of subsets of a given set is 2n
power set of A = 25 = 32 {Take [2n]}

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