To find the number of proper subsets of a set $S = \{1, 2, 3\}$, follow these steps:

Step 1: Total number of subsets

For a set with $n$ elements, the total number of subsets (including the empty set and the set itself) is given by: $2^n$ In this case, $n = 3$ (since the set $S$ has three elements), so the total number of subsets is: $2^3 = 8$

Step 2: Proper subsets

A proper subset is any subset of a set that is not equal to the set itself. To find the number of proper subsets, we subtract 1 (since the set itself is not considered a proper subset): $\text{Number of proper subsets} = 2^n - 1 = 8 - 1 = 7$

So, the number of proper subsets of the set $S = \{1, 2, 3\}$ is 7.

Would you like further details or have any questions?

Here are five related questions:

1. What are the proper subsets of the set $S = \{1, 2, 3\}$?
2. How many subsets does a set with 4 elements have?
3. What is the difference between a subset and a proper subset?
4. How does the formula for the number of subsets work for an empty set?
5. Can a set be a subset of itself?

Tip: The total number of subsets of a set doubles with each additional element because each element has two possibilities: being included or excluded.