Let N denote the set of natural numbers and `A = {n^(2) : a in N} andB= {n^(3) : n in N}`. Which one of the following is not correct?

Options

A. `A cup B = N`
B. The complement of `(A cup B)` is an infinite set
C. `A cap B` must be finite set
D. `A cap B` must be proper subset of `{m^(6) : m in n}`

Answer

To solve the problem, we need to analyze the sets \( A \) and \( B \) defined in the question and determine which statement about them is not correct.

### Step-by-Step Solution:

1. **Define the sets \( A \) and \( B \)**:
- The set \( A \) is defined as \( A = \{ n^2 : n \in N \} \). This means \( A \) contains all perfect squares of natural numbers. The elements of \( A \) are:
\[ A = \{ 1^2, 2^2, 3^2, 4^2, 5^2, \ldots \} = \{ 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, \ldots \} \]

- The set \( B \) is defined as \( B = \{ n^3 : n \in N \} \). This means \( B \) contains all perfect cubes of natural numbers. The elements of \( B \) are:
\[ B = \{ 1^3, 2^3, 3^3, 4^3, 5^3, \ldots \} = \{ 1, 8, 27, 64, 125, 216, 343, 512, \ldots \} \]

2. **Identify the elements of \( A \) and \( B \)**:
- From the definitions, we can see that both sets contain infinite elements. \( A \) consists of perfect squares, while \( B \) consists of perfect cubes.

3. **Consider the union \( A \cup B \)**:
- The union of sets \( A \) and \( B \) will include all elements that are either perfect squares or perfect cubes. Thus:
\[ A \cup B = \{ 1, 4, 8, 9, 16, 25, 27, 36, 49, 64, 81, 100, 125, \ldots \} \]
- Since both \( A \) and \( B \) are infinite, their union \( A \cup B \) is also infinite.

4. **Consider the intersection \( A \cap B \)**:
- The intersection of sets \( A \) and \( B \) consists of elements that are both perfect squares and perfect cubes. These are the perfect sixth powers:
\[ A \cap B = \{ n^6 : n \in N \} = \{ 1, 64, 729, \ldots \} \]
- This set is also infinite, as there are infinitely many natural numbers \( n \).

5. **Evaluate the statements**:
- **Statement A**: \( A \cup B = N \) (not correct, as there are natural numbers missing).
- **Statement B**: \( A \cup B \) is infinite (correct).
- **Statement C**: \( A \cap B \) is finite (not correct, as shown above).
- **Statement D**: \( A \cap B \) contains elements like \( n^6 \) (correct).

6. **Conclusion**:
- The statement that is **not correct** is **Statement A**: \( A \cup B \) does not equal the set of all natural numbers \( N \).

### Final Answer:
The statement that is not correct is **Statement A**: \( A \cup B \neq N \).