Write all the possible proper subsets of `{3,10}`

Answer

To find all the possible proper subsets of the set `{3, 10}`, we can follow these steps:

### Step 1: Identify the Set
We start with the set \( A = \{3, 10\} \).

### Step 2: Determine the Number of Elements
Count the number of elements in the set \( A \). Here, \( n = 2 \) (the elements are 3 and 10).

### Step 3: Calculate the Total Number of Subsets
Using the formula for the total number of subsets, which is \( 2^n \):
\[ 2^2 = 4 \]
This means there are 4 subsets in total.

### Step 4: List All Subsets
Now, we will list all the subsets of the set \( A \):
1. The empty set: \( \emptyset \)
2. The set containing the first element: \( \{3\} \)
3. The set containing the second element: \( \{10\} \)
4. The set containing both elements: \( \{3, 10\} \)

So, the subsets of \( A \) are:
\[ \{\emptyset, \{3\}, \{10\}, \{3, 10\}\} \]

### Step 5: Identify Proper Subsets
Proper subsets are all the subsets except the set itself and the empty set is not considered a proper subset. Therefore, the proper subsets of \( A \) are:
1. \( \{3\} \)
2. \( \{10\} \)

### Conclusion
The proper subsets of the set \( A = \{3, 10\} \) are:
\[ \{\{3\}, \{10\}\} \]

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