How many proper subsets in all are there of a set containing 3 elements ?

Answer

To find the number of proper subsets of a set containing 3 elements, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Definition of Proper Subset**:
A proper subset of a set A is any subset of A that is not equal to A itself. For example, if A = {1, 2, 3}, the proper subsets would include subsets like {1}, {2}, {3}, {1, 2}, {1, 3}, and {2, 3}, but not {1, 2, 3}.

**Hint**: Remember that a proper subset cannot be the entire set.

2. **Use the Formula for Total Subsets**:
The total number of subsets of a set containing n elements is given by the formula \(2^n\). This includes all possible subsets, including the empty set and the set itself.

**Hint**: Think of each element in the set as having two choices: to be included in the subset or not.

3. **Calculate Total Subsets for 3 Elements**:
For a set with 3 elements (n = 3), we can calculate the total number of subsets:
\[ 2^3 = 8 \]

**Hint**: Calculate \(2^3\) to find the total number of subsets.

4. **Subtract the Set Itself**:
Since we are looking for proper subsets, we need to exclude the set itself from our total count. Therefore, we subtract 1 from the total number of subsets:
\[ 8 - 1 = 7 \]

**Hint**: Remember to subtract the one subset that is the set itself.

5. **Conclusion**:
The number of proper subsets of a set containing 3 elements is 7.

**Final Answer**: 7