Which of the following pairs of sets are not equal ?

Options

A. `A = {1, 3, 3, 1} , B = {1. 4}`
B. `A = {x| x + 4 = 4 }, B = {0}`
C. `A = {m, a, t, h, e, i , c, s}, B = {a, m, t, h, e, i, c , s}`
D. `A = {1, 4, 9, 16, 25, . . . } B = {x|x= n^(2) , n in N}`

Answer

To determine which pairs of sets are not equal, we will analyze each option step by step.

### Step-by-Step Solution:

1. **Understanding Equal Sets**:
- Two sets are equal if they have the same elements, regardless of the order or repetition of those elements.

2. **Option 1**:
- Set A: {1, 3, 3, 1} → This simplifies to {1, 3} (removing duplicates).
- Set B: {1, 4}
- Comparison: Set A has elements 1 and 3, while Set B has elements 1 and 4. Since Set B contains the element 4, which is not in Set A, these sets are not equal.
- Conclusion: **Option 1 is not equal.**

3. **Option 2**:
- Set A: Defined by the equation \( x + 4 = 4 \).
- Solving gives \( x = 0 \), so Set A = {0}.
- Set B: Given as {0}.
- Comparison: Both sets contain the same element (0), hence they are equal.
- Conclusion: **Option 2 is equal.**

4. **Option 3**:
- Set A: {1, 4, 9, 16, 25}
- Set B: Defined by \( x = n^2 \) where \( n \) is a natural number.
- For \( n = 1, 2, 3, 4, 5 \), we get \( x = 1^2, 2^2, 3^2, 4^2, 5^2 \) which results in {1, 4, 9, 16, 25}.
- Comparison: Both sets contain the same elements {1, 4, 9, 16, 25}, hence they are equal.
- Conclusion: **Option 3 is equal.**

5. **Option 4**:
- Set A: {1, 4, 9, 16, 25} (same as in Option 3).
- Set B: Defined by \( x = n^2 \) for natural numbers \( n \).
- As previously calculated, Set B also results in {1, 4, 9, 16, 25}.
- Comparison: Both sets are the same, hence they are equal.
- Conclusion: **Option 4 is equal.**

### Final Conclusion:
- The only pair of sets that are not equal is found in **Option 1**.