If `A={(x,y):y=1/x, 0 != x in R}, B={(x,y):y=-x,x in R}` then (i) `AnnB=A` (ii) `AnnB=B` (iii) `AnnB=phi` (iv) `AuuB=A`

Options

A. `AnnB=A`
B. `AnnB=B`
C. `AnnB=phi`
D. `AuuB=A`

Answer

To solve the problem, we need to analyze the given sets A and B and determine their intersection and union.

### Step 1: Define the sets A and B
- Set A is defined as:
\[ A = \{(x, y) : y = \frac{1}{x}, x \neq 0, x \in \mathbb{R}\} \]
This means A contains all pairs (x, y) where y is the reciprocal of x, and x can take any real number except zero.

- Set B is defined as:
\[ B = \{(x, y) : y = -x, x \in \mathbb{R}\} \]
This means B contains all pairs (x, y) where y is the negative of x.

### Step 2: Find the intersection of sets A and B
To find \( A \cap B \), we need to see if there are any common elements between the two sets.

1. For a point \((x, y)\) to be in both A and B, it must satisfy both conditions:
- From A: \( y = \frac{1}{x} \)
- From B: \( y = -x \)

2. Set these two equations equal to each other:
\[ \frac{1}{x} = -x \]

3. Multiply both sides by \(x\) (keeping in mind \(x \neq 0\)):
\[ 1 = -x^2 \]
This implies:
\[ x^2 = -1 \]
Since there are no real solutions to this equation, it indicates that there are no common elements between sets A and B.

Thus, we conclude:
\[ A \cap B = \emptyset \quad \text{(denoted as } \phi\text{)} \]

### Step 3: Find the union of sets A and B
Next, we find \( A \cup B \):
- The union of two sets includes all elements that are in either set A or set B. Since there are no common elements, the union will simply be the combination of both sets.

Thus:
\[ A \cup B = A \text{ (since there are no elements in B that are also in A)} \]

### Conclusion
Based on the analysis:
- \( A \cap B = \phi \) (option iii is correct)
- \( A \cup B = A \) (option iv is also correct)

### Final Answer
The correct options are:
- (iii) \( A \cap B = \phi \)
- (iv) \( A \cup B = A \)