Are the following of sets equal ?
(i) `{x:x` is a letter in the word 'INDIA'} `{x:x` is letter in the word 'DINA'}.
(ii) `A={x:x` is a root of the equation `x^(2)-5x+6=0},B={2,3}.`
(iii) `A=:1,2,4,},B={x:x` is a dighit in the number `1123421}`
(iv) `A={2,3,5},B={x:x` is a positive prime divisor of `180}`
(v) `A={1,4,8,16,64},B={x:x` is a positive divisor of `64}.`

Answer

To determine if the given sets are equal, we will analyze each pair of sets step by step.

### (i) Sets:
- \( A = \{x : x \text{ is a letter in the word 'INDIA'}\} \)
- \( B = \{x : x \text{ is a letter in the word 'DINA'}\} \)

**Step 1:** Identify the unique letters in 'INDIA'.
- The letters are I, N, D, A.
- Thus, \( A = \{I, N, D, A\} \).

**Step 2:** Identify the unique letters in 'DINA'.
- The letters are D, I, N, A.
- Thus, \( B = \{D, I, N, A\} \).

**Step 3:** Compare the sets.
- Both \( A \) and \( B \) contain the same elements: \( \{I, N, D, A\} \) and \( \{D, I, N, A\} \).
- Therefore, \( A = B \).

### (ii) Sets:
- \( A = \{x : x \text{ is a root of the equation } x^2 - 5x + 6 = 0\} \)
- \( B = \{2, 3\} \)

**Step 1:** Solve the quadratic equation \( x^2 - 5x + 6 = 0 \).
- Factor the equation: \( (x - 2)(x - 3) = 0 \).
- The roots are \( x = 2 \) and \( x = 3 \).
- Thus, \( A = \{2, 3\} \).

**Step 2:** Compare the sets.
- Since \( A = \{2, 3\} \) and \( B = \{2, 3\} \), we have \( A = B \).

### (iii) Sets:
- \( A = \{1, 2, 4\} \)
- \( B = \{x : x \text{ is a digit in the number } 1123421\} \)

**Step 1:** Identify the unique digits in 1123421.
- The digits are 1, 2, 3, 4.
- Thus, \( B = \{1, 2, 3, 4\} \).

**Step 2:** Compare the sets.
- \( A = \{1, 2, 4\} \) and \( B = \{1, 2, 3, 4\} \).
- Since \( B \) has an additional element (3), \( A \neq B \).

### (iv) Sets:
- \( A = \{2, 3, 5\} \)
- \( B = \{x : x \text{ is a positive prime divisor of } 180\} \)

**Step 1:** Find the positive prime divisors of 180.
- The prime factorization of 180 is \( 2^2 \times 3^2 \times 5 \).
- The positive prime divisors are 2, 3, and 5.
- Thus, \( B = \{2, 3, 5\} \).

**Step 2:** Compare the sets.
- Since \( A = \{2, 3, 5\} \) and \( B = \{2, 3, 5\} \), we have \( A = B \).

### (v) Sets:
- \( A = \{1, 4, 8, 16, 64\} \)
- \( B = \{x : x \text{ is a positive divisor of } 64\} \)

**Step 1:** Identify the positive divisors of 64.
- The positive divisors of 64 are 1, 2, 4, 8, 16, 32, 64.
- Thus, \( B = \{1, 2, 4, 8, 16, 32, 64\} \).

**Step 2:** Compare the sets.
- \( A = \{1, 4, 8, 16, 64\} \) and \( B = \{1, 2, 4, 8, 16, 32, 64\} \).
- Since \( B \) has additional elements (2 and 32), \( A \neq B \).

### Summary of Results:
1. \( A = B \) (for the first set)
2. \( A = B \) (for the second set)
3. \( A \neq B \) (for the third set)
4. \( A = B \) (for the fourth set)
5. \( A \neq B \) (for the fifth set)