X, Y and Z are closed intervals of unit length on the real line. The overlap…
2000
X, Y and Z are closed intervals of unit length on the real line. The overlap of X and Y is half a unit. The overlap of Y and Z is also half a unit. Let the overlap of X and Z be k units. Which of the following is true?
- A.
k must be 1
- B.
k must be 0
- C.
k can take any value between 0 and 1
- D.
None of the above
Attempted by 36 students.
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Correct answer: D
The correct answer is: None of the above.
Let Y = [0, 1]. A unit interval overlaps Y by half a unit exactly when its starting point is shifted by 0.5 to the left or right of Y.
Therefore X can be [-0.5, 0.5] or [0.5, 1.5]. Similarly, Z can be [-0.5, 0.5] or [0.5, 1.5].
If X and Z are on the same side of Y, they coincide, so k = 1. If they are on opposite sides, they only meet at an endpoint, so the overlap length is k = 0.
Thus k is not forced to be 0 or 1, and it cannot take every value between 0 and 1. Only k = 0 or k = 1 are possible, so none of the first three statements is true.