If `A={x in CC : x^(4) -1=0}`
`B={x in CC :x^(2) -1=0}`
`C={x in CC :x^(2)+1=0}`
Where `CC` is complex plane.

Options

A. `A=B cup C`
B. `C=A cap B`
C. `B=A cap C`
D. `A=B cap C`

Answer

To solve the problem, we need to find the elements of the sets A, B, and C defined in the complex plane (CC).

### Step 1: Find the elements of set A
Set A is defined as:
\[ A = \{ x \in \mathbb{C} : x^4 - 1 = 0 \} \]

To find the roots of the equation \( x^4 - 1 = 0 \), we can factor it as follows:
\[ x^4 - 1 = (x^2 - 1)(x^2 + 1) = 0 \]

Now, we can solve each factor separately:
1. \( x^2 - 1 = 0 \)
- This gives us \( x^2 = 1 \), so \( x = 1 \) or \( x = -1 \).

2. \( x^2 + 1 = 0 \)
- This gives us \( x^2 = -1 \), so \( x = i \) or \( x = -i \) (where \( i \) is the imaginary unit).

Thus, the elements of set A are:
\[ A = \{ 1, -1, i, -i \} \]

### Step 2: Find the elements of set B
Set B is defined as:
\[ B = \{ x \in \mathbb{C} : x^2 - 1 = 0 \} \]

To find the roots of the equation \( x^2 - 1 = 0 \):
- This gives us \( x^2 = 1 \), so \( x = 1 \) or \( x = -1 \).

Thus, the elements of set B are:
\[ B = \{ 1, -1 \} \]

### Step 3: Find the elements of set C
Set C is defined as:
\[ C = \{ x \in \mathbb{C} : x^2 + 1 = 0 \} \]

To find the roots of the equation \( x^2 + 1 = 0 \):
- This gives us \( x^2 = -1 \), so \( x = i \) or \( x = -i \).

Thus, the elements of set C are:
\[ C = \{ i, -i \} \]

### Step 4: Determine the relationship between sets A, B, and C
Now we can analyze the relationships between the sets:
- Set A contains all elements from both sets B and C.
- Specifically, \( A = B \cup C \).

Thus, we conclude that:
\[ A = \{ 1, -1, i, -i \} = B \cup C \]

### Final Answer
The relationship between the sets is:
\[ A = B \cup C \]