If the sets A and B are defined as A = {(x, y) | y = 1 / x, 0 ≠ x ∈ R}, B = {(x, y)|y = -x ∈ R} then

Sets: \(A=\{(x,y)\mid y=\tfrac{1}{x},\ x\in\mathbb{R}\setminus\{0\}\}\), \(B=\{(x,y)\mid y=-x,\ x\in\mathbb{R}\}\).

Intersection: Solve \( \frac{1}{x} = -x \) with \(x\neq 0\). \[ \frac{1}{x} = -x \;\Longrightarrow\; 1 = -x^2 \;\Longrightarrow\; x^2 = -1, \] which has no real solution.

Conclusion: \(A \cap B = \varnothing\) (they are disjoint in \(\mathbb{R}^2\)).

Note: Over complex numbers, the intersection would be at \(x=\pm i\), but for real \(x\), there is none.