If `A={x : -3 lt x lt 3, x in Z}` then find the number of subsets of `A` .

Answer

To solve the problem, we need to find the set \( A \) defined by the condition \( -3 < x < 3 \) where \( x \) is an integer (i.e., \( x \in \mathbb{Z} \)). Then, we will determine the number of subsets of this set \( A \).

### Step-by-Step Solution:

1. **Identify the Set \( A \)**:
- The condition \( -3 < x < 3 \) implies that \( x \) can take integer values between -3 and 3, but not including -3 and 3.
- Therefore, the integer values that satisfy this condition are:
\[ -2, -1, 0, 1, 2 \]
- Thus, we can write:
\[ A = \{-2, -1, 0, 1, 2\} \]

2. **Count the Elements in Set \( A \)**:
- The elements of set \( A \) are:
\[ -2, -1, 0, 1, 2 \]
- There are a total of 5 elements in set \( A \).
- Therefore, we have:
\[ |A| = 5 \]

3. **Calculate the Number of Subsets**:
- The number of subsets of a set is given by the formula \( 2^{|A|} \), where \( |A| \) is the number of elements in the set.
- Since we found that \( |A| = 5 \), we can calculate the number of subsets:
\[ \text{Number of subsets} = 2^{|A|} = 2^5 = 32 \]

4. **Conclusion**:
- The number of subsets of the set \( A \) is:
\[ \boxed{32} \]