X, Y, and Z are closed intervals of unit length on the real line. The length…
2000
X, Y, and Z are closed intervals of unit length on the real line. The length of the overlap of X and Y is half a unit, and the length of the overlap of Y and Z is also half a unit. Let k denote the length of the overlap of X and Z. Which statement is true?
Answer: D. None of these statements — ConceptFor two closed intervals of equal length 1 whose left endpoints differ by d, the length of their overlap is max(0, 1 − |d|). Endpoint contact has…
- A.
k must equal 1
- B.
k must equal 0
- C.
k can be any real number in [0, 1]
- D.
None of these statements
Attempted by 117 students.
Show answer & explanation
Correct answer: D
Concept
For two closed intervals of equal length 1 whose left endpoints differ by d, the length of their overlap is max(0, 1 − |d|). Endpoint contact has overlap length 0.
Thus an overlap length of 1/2 forces the left endpoints to differ by exactly 1/2; there are two possible directions for that shift.
Application
Set Y = [0, 1]. Since X overlaps Y by 1/2, X is either [−1/2, 1/2] or [1/2, 3/2].
The same reasoning gives Z as either [−1/2, 1/2] or [1/2, 3/2].
If X and Z use the same shift, they coincide, so their overlap length is k = 1.
If X and Z use opposite shifts, they meet only at the endpoint 1/2, so their overlap length is k = 0.
Cross-check and result
Placement of X | Placement of Z | Overlap length k |
|---|---|---|
[−1/2, 1/2] | [−1/2, 1/2] | 1 |
[−1/2, 1/2] | [1/2, 3/2] | 0 |
[1/2, 3/2] | [−1/2, 1/2] | 0 |
[1/2, 3/2] | [1/2, 3/2] | 1 |
The four ordered placement pairs produce overlap lengths 1, 0, 0, and 1. No intermediate value occurs.
Therefore k can be only 0 or 1: it is not forced to either single value, and it cannot be every real number in [0, 1]. Hence the true statement is ‘None of these statements.’
Explore the full course: Iocl Engineers Officers Grade A Paper 2