Number of Subsets of a given Set:
If a set contains ‘n’ elements, then the number of subsets of the set is 2\(^{2}\).
Number of Proper Subsets of the Set:
If a set contains ‘n’ elements, then the number of proper subsets of the set is 2\(^{n}\)  1.
If A = {p, q} the proper subsets of A are [{ }, {p}, {q}]
⇒ Number of proper subsets of A are 3 = 2\(^{2}\)  1 = 4  1
In general, number of proper subsets of a given set = 2\(^{m}\)  1, where m is the number of elements.
For example:
1. If A {1, 3, 5}, then write all the possible subsets of A. Find their numbers.
Solution:
The subset of A containing no elements  { }
The subset of A containing one element each  {1} {3} {5}
The subset of A containing two elements each  {1, 3} {1, 5} {3, 5}
The subset of A containing three elements  {1, 3, 5)
Therefore, all possible subsets of A are { }, {1}, {3}, {5}, {1, 3}, {3, 5}, {1, 3, 5}
Therefore, number of all possible subsets of A is 8 which is equal 2\(^{3}\).
Proper subsets are = { }, {1}, {3}, {5}, {1, 3}, {3, 5}
Number of proper subsets are 7 = 8  1 = 2\(^{3}\)  1
2. If the number of elements in a set is 2, find the number of subsets and proper subsets.
Solution:
Number of elements in a set = 2
Then, number of subsets = 2\(^{2}\) = 4
Also, the number of proper subsets = 2\(^{2}\)  1
= 4 – 1 = 3
3. If A = {1, 2, 3, 4, 5}
then the number of proper subsets = 2\(^{5}\)  1
= 32  1 = 31 {Take [2\(^{n}\)  1]}
and
power set of A = 2\(^{5}\) = 32 {Take [2\(^{n}\)]}
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