Question

If $A=\{1,2,3,4,5\}$, then the number of proper subsets of $A$ is
A. $31$
B. $38$
C. $48$
D. $54$

Answer 1

Hint: To solve this question we will use the formula of calculating number of proper subsets from the given set $A$. We will first calculate the value of $n$ by determining the number of elements in the given set then substitute it in the formula and find the number of proper subsets.

Formula Used: Number of proper subsets $={{2}^{n}}-1$ where $n$ is the number of elements.

Complete step by step solution: We Have given a set of values $A=\{1,2,3,4,5\}$ and we have to find the number of proper subsets of $A$. We will use the formula of calculating number of proper subsets ${{2}^{n}}-1$ . Now here in set $A$ we are given five elements so value of $n$ will be $n=5$.
Number of proper subsets will be,
$\begin{align}
  & ={{2}^{n}}-1 \\
 & ={{2}^{5}}-1 \\
 & =32-1 \\
 & =31 \\
\end{align}$
The total number of proper subsets of $A=\{1,2,3,4,5\}$ is $31$.

Option ‘A’ is correct

Note: A set can be defined as the collection of elements or any objects which is grouped and enclosed in a curly bracket whereas subset is a part of a set which means that it can contain all, none or some of the elements from the parent set.
When a subset contains some of the elements from a parent set that is it must contain at least one element from the set then that subset is termed as proper subset. It is denoted with symbol $\subset $. If a subset contains all the elements from the parent set, then it is termed as improper set.