Number of proper subsets of a set containing 4 elements is
(i) `4^(2)`
(ii) `4^(2)-1`
(iii) `2^(4)`
(iv) `2^(4)-1`

Options

A. `4^(2)`
B. `4^(2)-1`
C. `2^(4)`
D. `2^(4)-1`

Answer

To find the number of proper subsets of a set containing 4 elements, we can follow these steps:

### Step 1: Understand the definition of subsets
A subset is a set that contains some or all elements of another set. A proper subset is a subset that contains at least one element but is not equal to the original set.

### Step 2: Determine the number of elements in the set
In this case, we have a set containing 4 elements. Let's denote the number of elements as \( n = 4 \).

### Step 3: Use the formula for the number of subsets
The total number of subsets of a set with \( n \) elements is given by the formula:
\[ 2^n \]
This includes both proper and improper subsets (the set itself).

### Step 4: Calculate the total number of subsets
For our set with 4 elements, the total number of subsets is:
\[ 2^4 = 16 \]

### Step 5: Calculate the number of proper subsets
The number of proper subsets is given by the total number of subsets minus the set itself:
\[ \text{Number of proper subsets} = 2^n - 1 \]
Substituting \( n = 4 \):
\[ \text{Number of proper subsets} = 2^4 - 1 = 16 - 1 = 15 \]

### Conclusion
The number of proper subsets of a set containing 4 elements is \( 15 \).

### Answer Selection
Now, let's check the options provided:
(i) \( 4^2 \) = 16 (not correct)
(ii) \( 4^2 - 1 \) = 15 (correct)
(iii) \( 2^4 \) = 16 (not correct)
(iv) \( 2^4 - 1 \) = 15 (also correct)

Both options (ii) and (iv) yield the correct number of proper subsets, which is 15.