Introduction / Context:
We must uniquely identify the shortest person using partial ranking facts.
Given Data / Assumptions:
- Six distinct heights: A, B, C, D, E, F.
- I: Only B is taller than C ⇒ B is tallest, C second tallest.
- II: A is taller than only D and F ⇒ D and F are both shorter than A; E is unconstrained.
Concept / Approach:
Combine partial orders and see if the minimum (shortest) is forced.
Step-by-Step Solution:
From I alone: Shortest could be any of A, D, E, F depending on unknowns. Not sufficient.From II alone: Among {A,D,F}, either D or F could be the shortest (A is above them), but E's position relative to D/F is unknown; shortest remains ambiguous.Combine I & II: We now know the top two (B, C) and that A is above D and F. E can be anywhere below C and potentially below A as well. The absolute minimum could be D or F or even E, depending on actual heights. Thus, shortest is not uniquely determined.
Verification / Alternative check:
Construct examples: Case 1 shortest=F (F<D<A<E<C<B). Case 2 shortest=E (E<F<D<A<C<B). Both satisfy I and II.
Why Other Options Are Wrong:
- I alone / II alone: each leaves multiple candidates for shortest.
- Either alone sufficient / Both together sufficient: contradicted by counterexamples.
Common Pitfalls:
Assuming E's position without data; assuming D and F order from II (not given).
Final Answer:
Both Statements I and II together are not sufficient.
Discussion & Comments