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Difficulty: HardSufficient Assumption

Astrophysicist: Any star cluster that formed in an environment with high stellar density underwent rapid gravitational relaxation during its infancy. Star clusters that undergo rapid gravitational relaxation inevitably expel all of their intermediate-mass black holes through stellar dynamical interactions. The Pyxis cluster currently contains several intermediate-mass black holes. Therefore, the Pyxis cluster must have originated in a region with abundant molecular gas clouds.

The astrophysicist's conclusion follows logically if which one of the following is assumed?

  1. A
    Every star cluster that originated in a region with abundant molecular gas clouds did not form in an environment with high stellar density.
  2. B
    The Pyxis cluster did not undergo rapid gravitational relaxation during its infancy.
  3. Any star cluster that did not originate in a region with abundant molecular gas clouds formed in an environment with high stellar density.Answer
  4. D
    A star cluster can originate in a region with abundant molecular gas clouds only if it retains its intermediate-mass black holes.
  5. E
    All star clusters that retain intermediate-mass black holes completely avoid any gravitational interactions during their formation.

Answer

The assumption that any star cluster that did not originate in a region with abundant molecular gas clouds formed in an environment with high stellar density.
The credited response establishes the conditional rule that not originating in abundant molecular gas clouds implies forming in high stellar density (¬M→H\neg M \rightarrow H). The contrapositive of this statement is that not forming in high stellar density guarantees originating in abundant molecular gas clouds (¬H→M\neg H \rightarrow M). Because the premises prove that the Pyxis cluster did not form in high stellar density (¬H\neg H), this rule rigorously guarantees that Pyxis must have originated in a region with abundant molecular gas clouds.

Step-by-Step Solution

1
Analyze and diagram the formal conditional relationships in the premises.
Premise 1: Formed in high stellar density →\rightarrow Rapid gravitational relaxation (H→RH \rightarrow R). Premise 2: Rapid gravitational relaxation →\rightarrow Expel all intermediate-mass black holes (R→No BR \rightarrow \text{No } B). Combining these yields H→R→No BH \rightarrow R \rightarrow \text{No } B, with the contrapositive being: Contains intermediate-mass black holes →\rightarrow Did not undergo rapid gravitational relaxation →\rightarrow Did not form in high stellar density (B→¬R→¬HB \rightarrow \neg R \rightarrow \neg H).
Deconstructing the conditional chain isolates what is already proven from what remains unproven.
2
Evaluate the specific evidence given about the subject and identify the inferential gap.
The Pyxis cluster contains intermediate-mass black holes (BPyxisB_{\text{Pyxis}}), which deductively proves that Pyxis did not form in high stellar density (¬HPyxis\neg H_{\text{Pyxis}}). The conclusion claims that Pyxis originated in a region with abundant molecular gas clouds (MPyxisM_{\text{Pyxis}}). The deductive gap is between ¬HPyxis\neg H_{\text{Pyxis}} and MPyxisM_{\text{Pyxis}}.
A sufficient assumption must provide a conditional rule where the proven fact (¬H\neg H) guarantees the final conclusion (MM).
3
Identify the missing sufficient bridge condition.
We need the rule ¬H→M\neg H \rightarrow M, which is logically equivalent to its contrapositive ¬M→H\neg M \rightarrow H ('If a star cluster did not originate in abundant molecular gas clouds, it formed in high stellar density').
Adding ¬H→M\neg H \rightarrow M to the premise deduction ¬HPyxis\neg H_{\text{Pyxis}} 100% logically forces the conclusion MPyxisM_{\text{Pyxis}}.

Key Concept

Sufficient Assumption / Conditional Contrapositive Chain
Estimated Time:2m 0s
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Topics

Sufficient Assumption

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