Write the following sets in roster form and also in set builder form :
B = {two digit numbers which are perfect square}

Answer

To solve the problem of writing the set of two-digit numbers which are perfect squares in both roster form and set builder form, we will follow these steps:

### Step 1: Identify the range of two-digit numbers
Two-digit numbers range from 10 to 99.

### Step 2: Find the perfect squares within this range
We need to find the perfect squares of natural numbers that fall within the range of two-digit numbers.

- The smallest two-digit number is 10. The smallest integer whose square is a two-digit number is 4, since \(4^2 = 16\).
- The largest two-digit number is 99. The largest integer whose square is a two-digit number is 9, since \(9^2 = 81\).

### Step 3: Calculate the perfect squares
Now we calculate the perfect squares of integers from 4 to 9:

- \(4^2 = 16\)
- \(5^2 = 25\)
- \(6^2 = 36\)
- \(7^2 = 49\)
- \(8^2 = 64\)
- \(9^2 = 81\)

### Step 4: Write the roster form
The roster form of the set includes all the elements we found:

**Roster Form:**
\[ B = \{16, 25, 36, 49, 64, 81\} \]

### Step 5: Write the set builder form
In set builder form, we describe the set using a variable and a condition.

**Set Builder Form:**
\[ B = \{ x \mid x = n^2, \, n \in \mathbb{N}, \, 10 \leq x \leq 99 \} \]

Where \( n \) is a natural number.

### Summary of the Solution
- Roster Form: \( B = \{16, 25, 36, 49, 64, 81\} \)
- Set Builder Form: \( B = \{ x \mid x = n^2, \, n \in \mathbb{N}, \, 10 \leq x \leq 99 \} \)

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