If A = {1, 2, 3, 4, 5), then the number of proper subsets of A is

Options

A. a) 31
B. b) 38
C. c) 48
D. d) 54

Answer

To find the number of proper subsets of the set \( A = \{1, 2, 3, 4, 5\} \), we can follow these steps:

### Step 1: Identify the number of elements in set A
The set \( A \) contains the elements \( 1, 2, 3, 4, 5 \). We can count the elements:
- Number of elements in \( A \) (denoted as \( n \)) = 5.

### Step 2: Calculate the total number of subsets
The formula for the total number of subsets of a set is given by \( 2^n \), where \( n \) is the number of elements in the set.
- Here, \( n = 5 \), so the total number of subsets is:
\[ 2^5 = 32. \]

### Step 3: Calculate the number of proper subsets
A proper subset of a set is any subset that is not equal to the set itself. Thus, to find the number of proper subsets, we subtract 1 from the total number of subsets (to exclude the set \( A \) itself).
- Therefore, the number of proper subsets is:
\[ 2^5 - 1 = 32 - 1 = 31. \]

### Conclusion
The number of proper subsets of the set \( A \) is \( 31 \).

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