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Zorluk: OrtaIdentifying Must-Be-True Statements and Valid Deductions

Under the revised regulations of the Global Mineral Oversight Commission, any rare-earth mining facility operating in Zone 4 that extracts dysprosium is required to undergo a quarterly heavy-metals soil audit. Furthermore, every mining facility that undergoes a quarterly heavy-metals soil audit is automatically awarded an Eco-Compliance Certificate by the commission. Facility Alpha operates in Zone 4, but it has not been awarded an Eco-Compliance Certificate by the commission.

If the statements above are true, which of the following must also be true based on them?

  1. Facility Alpha does not extract dysprosium.Cevap
  2. B
    Facility Alpha underwent a quarterly heavy-metals soil audit but failed to meet the required environmental safety standards.
  3. C
    If a mining facility has been awarded an Eco-Compliance Certificate, that facility must extract dysprosium.
  4. D
    Facility Alpha extracts rare-earth elements other than dysprosium at its Zone 4 location.
  5. E
    No mining facility operating outside of Zone 4 is required to undergo a quarterly heavy-metals soil audit.

Cevap

Facility Alpha does not extract dysprosium.
Combining the premises yields the conditional rule: if a facility operates in Zone 4 and extracts dysprosium, it receives an Eco-Compliance Certificate. The contrapositive dictates that if a facility lacks an Eco-Compliance Certificate, it either does not operate in Zone 4 or does not extract dysprosium. Because the stimulus explicitly states that Facility Alpha does operate in Zone 4, it must logically follow that Facility Alpha does not extract dysprosium.

Adım Adım Çözüm

1
Analyze the conditional premises provided in the stimulus.
Premise 1: (Operates in Zone 4 AND Extracts Dysprosium) → Undergoes quarterly soil audit. Premise 2: Undergoes quarterly soil audit → Awarded Eco-Compliance Certificate.
Establishing conditional relationships allows for logical chaining.
2
Chain the premises together to form a combined conditional statement.
(Operates in Zone 4 AND Extracts Dysprosium) → Awarded Eco-Compliance Certificate.
If A leads to B, and B leads to C, then A directly leads to C.
3
Formulate the contrapositive of the combined conditional statement.
NOT Awarded Eco-Compliance Certificate → NOT (Operates in Zone 4 AND Extracts Dysprosium) = (Does NOT operate in Zone 4) OR (Does NOT extract Dysprosium).
The contrapositive of any valid conditional statement is logically equivalent and must also be true.
4
Apply the facts about Facility Alpha to the contrapositive.
Facility Alpha has NOT been awarded an Eco-Compliance Certificate. Since Facility Alpha IS operating in Zone 4, the second condition of the OR clause must hold: Facility Alpha does NOT extract dysprosium.
Since the first disjunct (does not operate in Zone 4) is false, the second disjunct must be true for the contrapositive to hold.

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Identifying Must-Be-True Statements and Valid Deductions
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Identifying Must-Be-True Statements and Valid Deductions

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