List all the proper subsets of `{0,{1},3}`.

Answer

To find all the proper subsets of the set \( A = \{0, \{1\}, 3\} \), we will follow these steps:

### Step 1: Identify the elements of the set
The set \( A \) contains three elements: \( 0 \), \( \{1\} \), and \( 3 \).

### Step 2: Calculate the total number of subsets
The formula to calculate the total number of subsets of a set with \( n \) elements is \( 2^n \). Here, \( n = 3 \) (since there are three elements in set \( A \)):
\[ 2^3 = 8 \]
Thus, there are 8 subsets in total.

### Step 3: List all the subsets
Now, we will list all the subsets of set \( A \):
1. The empty set: \( \emptyset \)
2. Single-element subsets:
- \( \{0\} \)
- \( \{\{1\}\} \)
- \( \{3\} \)
3. Two-element subsets:
- \( \{0, \{1\}\} \)
- \( \{0, 3\} \)
- \( \{\{1\}, 3\} \)
4. The set itself:
- \( \{0, \{1\}, 3\} \)

### Step 4: Identify the proper subsets
Proper subsets are all subsets except the set itself. Therefore, we will exclude \( \{0, \{1\}, 3\} \) from our list. The proper subsets of \( A \) are:
1. \( \emptyset \)
2. \( \{0\} \)
3. \( \{\{1\}\} \)
4. \( \{3\} \)
5. \( \{0, \{1\}\} \)
6. \( \{0, 3\} \)
7. \( \{\{1\}, 3\} \)

### Final Answer
The proper subsets of the set \( A = \{0, \{1\}, 3\} \) are:
1. \( \emptyset \)
2. \( \{0\} \)
3. \( \{\{1\}\} \)
4. \( \{3\} \)
5. \( \{0, \{1\}\} \)
6. \( \{0, 3\} \)
7. \( \{\{1\}, 3\} \)

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