Step-by-step Solution:

1. Define the Sets: \[ \text{Set } A \text{ consists of all points on the rectangular hyperbola } xy = 1. \] \[ \text{Set } B \text{ consists of all points on the line } y = -x. \] 2. Graphical Representation: \[ \text{The hyperbola } xy = 1 \text{ has branches in the first (I) and third (III) quadrants.} \] \[ \text{The line } y = -x \text{ has a slope of } -1 \text{ and passes through the origin.} \] 3. Intersection of Sets \( A \) and \( B \): To find the intersection \( A \cap B \), we solve the system of equations: \[ \begin{cases} xy = 1 \\ y = -x \end{cases} \] Substituting \( y = -x \) into \( xy = 1 \): \[ x(-x) = 1 \] \[ - x^2 = 1 \] \[ x^2 = -1 \] Since \( x^2 = -1 \) has no real solutions, the hyperbola and the line do not intersect. 4. Conclusion: \[ \text{Since there are no real points that satisfy both equations simultaneously, the intersection of sets } A \text{ and } B \text{ is the empty set.} \] \[ A \cap B = \emptyset \] Therefore, the intersection of sets \( A \) and \( B \) is: \[ \boxed{A \cap B = \emptyset} \]