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 follow these steps:

### Step 1: Identify the set A
We are given the set \( A = \{ x : -3 < x < 3, x \in \mathbb{Z} \} \). This means we need to find all integer values of \( x \) that lie between -3 and 3.

### Step 2: List the elements of set A
The integers that satisfy the condition \( -3 < x < 3 \) are:
- \( -2 \)
- \( -1 \)
- \( 0 \)
- \( 1 \)
- \( 2 \)

Thus, we can write:
\[ A = \{-2, -1, 0, 1, 2\} \]

### Step 3: Count the number of elements in set A
Now, we count the number of elements in set A. We have:
\[ |A| = 5 \]
where \( |A| \) denotes the cardinality (number of elements) of set A.

### Step 4: Calculate the number of subsets of A
The number of subsets of a set is given by the formula \( 2^{n} \), where \( n \) is the number of elements in the set. Therefore, the number of subsets of set A is:
\[ 2^{|A|} = 2^{5} = 32 \]

### Final Answer
The total number of subsets of the set \( A \) is \( 32 \).

---