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Difficulty: MediumDrawing Inferences and Valid Conclusions

A marine biology survey cataloged deep-sea coral reefs off the coast of Peninsula Y. The survey established that every reef located below a depth of 500 meters contains bioluminescent coral species. Furthermore, all coral reefs containing bioluminescent species in this region also host distinct populations of deep-sea shrimp. However, during the survey, several reefs located above a depth of 500 meters were found to host these same deep-sea shrimp populations.

Based on the survey results described above, which of the following statements must be true? (Select all that apply.)

  1. Every coral reef off Peninsula Y located below a depth of 500 meters hosts deep-sea shrimp populations.Answer
  2. Any coral reef off Peninsula Y that lacks deep-sea shrimp populations is located at a depth of 500 meters or less.Answer
  3. C
    Any coral reef off Peninsula Y that hosts deep-sea shrimp populations must also contain bioluminescent coral species.
  4. D
    Reefs located above a depth of 500 meters contain a larger total population of deep-sea shrimp than reefs located below 500 meters.
  5. E
    Water temperature variations between depth zones explain why deep-sea shrimp inhabit certain reefs located above 500 meters.

Answer

The statements establishing that every reef located below a depth of 500 meters hosts deep-sea shrimp populations, and that any reef lacking deep-sea shrimp populations must be located at a depth of 500 meters or less, are both valid deductions that must be true.
The passage establishes two linked conditional rules: any reef deeper than 500 meters contains bioluminescent coral, and any reef with bioluminescent coral hosts deep-sea shrimp. Combining these yields the direct deduction that every reef below 500 meters hosts deep-sea shrimp. Furthermore, taking the contrapositive of this combined rule shows that any reef lacking deep-sea shrimp cannot be below 500 meters, meaning it must be located at a depth of 500 meters or less. Both of these statements are logically necessary conclusions.

Step-by-Step Solution

1
Identify and map explicit conditional premises
Premise 1: Reef Depth > 500m → Bioluminescent Coral. Premise 2: Bioluminescent Coral → Deep-Sea Shrimp. Premise 3: Some Reefs Depth ≤ 500m → Deep-Sea Shrimp.
Mapping the statements into formal conditional logic clarifies valid inferential paths.
2
Synthesize the conditional premises into a transitive chain
Transitive chain: Reef Depth > 500m → Bioluminescent Coral → Deep-Sea Shrimp. Thus, Reef Depth > 500m → Deep-Sea Shrimp.
Combining the first two premises establishes that all reefs deeper than 500 meters necessarily host deep-sea shrimp.
3
Formulate the logically equivalent contrapositive of the transitive chain
Contrapositive: No Deep-Sea Shrimp → Reef Depth ≤ 500m.
If a conditional statement 'If X then Y' is true, its contrapositive 'If not Y then not X' is guaranteed to be true.

Key Concept

Conditional Logic & Valid Deductive Inferences
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Topics

Drawing Inferences and Valid Conclusions

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