Question

What is the total number of proper subsets of a set consisting of n elements?

Answer 1

Hint: If there exists a single element in set B that is not in set A then set B will not be considered as the subset of set A. A set B is known as the proper subset of set A if every element of set B is in set A but there exists at least any one element in set A that is not in set B.

Complete step-by-step solution:
Before moving forward, let us first understand what a set is. So the set is defined as the well-defined collection of objects. For example, the collection of all the vowels in English alphabets is a well-defined set, the collection of all rivers in India is also a well-defined set, etc. But the collection of all talented writers of India is not considered as a set because it is not well defined that all are talented writers.
Every set is considered as the subset of itself and the empty set is the subset of every set. The intersection of set A and set B contains all the elements which are common in both sets. The union of set A and set B contains all the elements which are either in set A, in set B, or in both of them.
If there are two sets A and B, such that all the elements of set B are in set A then, set B is known as the subset of set A.
\[B\subseteq A\]
If A and B are two sets such that B is the subset of A but A and B are not equal then, B is known as the proper subset of set A.
If B is the proper subset of set A, then
\[B\subseteq A\] but \[B\ne A\]
If we have n elements in the set then a total number of the subset that can be formed is shown below.
\[total\text{ }no.of\text{ }subset={{2}^{n}}\]
Suppose if we have n elements in our set then the total number of the proper subset that can be formed will be given as shown below.
\[total\text{ }no.of\text{ }proper\text{ }subset={{2}^{n}}-1\]

Note: A set can be represented in two forms. The first form is the roaster form and the second form is the set builder form. A set that does not contain any elements in it is termed an empty set or a null set. Two sets are said to be equivalent if the number of elements in each set comes out to be equal.