Let me find the least and greatest distances between points in sets S and T.
Set \(S = \{(x,y) : x \in \{-5,-4,-3,-2\}, y \in \{-3,-2\}\}\)
Set \(T = \{(x,y) : x \in \{2,3\}, y \in \{-3,-2,-1,0,1,2,3,4\}\}\)
For the least distance, I need to find the closest points between the two sets. Since the sets are separated horizontally, the minimum distance will occur between points with the same y-coordinate and the closest x-coordinates.
The rightmost x-value in S is -2 and the leftmost x-value in T is 2. Both sets contain y-values -3 and -2.
Closest points: \((-2, -3)\) in S and \((2, -3)\) in T (or \((-2, -2)\) and \((2, -2)\))
Distance = \(\sqrt{[(2-(-2))^2 + (-3-(-3))^2]} = \sqrt{[4^2 + 0^2]} = \sqrt{16} = 4\)
For the greatest distance, I need the farthest points, which will be at opposite corners of the rectangular regions.
Farthest points: \((-5, -3)\) in S and \((3, 4)\) in T
Distance = \(\sqrt{[(3-(-5))^2 + (4-(-3))^2]} = \sqrt{[8^2 + 7^2]} = \sqrt{[64 + 49]} = \sqrt{113}\)
Therefore, the least distance is 4 and the greatest distance is \(\sqrt{113}\).
