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Difficulty: MediumCannot Be True Questions

Under the new municipal regulations, any local business that fails the annual safety inspection is heavily fined and subjected to a mandatory operational audit. Some local businesses that are subjected to a mandatory operational audit receive municipal tax grants. However, no local business that receives a municipal tax grant has ever been heavily fined.

If the statements above are true, which one of the following CANNOT be true?

  1. A
    Any local business that is heavily fined failed the annual safety inspection.
  2. B
    A local business that passes the annual safety inspection is not subjected to a mandatory operational audit.
  3. A local business that receives a municipal tax grant failed the annual safety inspection.Answer
  4. D
    Most local businesses that receive municipal tax grants are not subjected to a mandatory operational audit.
  5. E
    Some local businesses that are subjected to a mandatory operational audit have never been heavily fined.

Answer

The statement that a local business receiving a municipal tax grant failed the annual safety inspection cannot be true.
Based on the stimulus, failing the safety inspection guarantees a heavy fine. However, receiving a municipal tax grant guarantees that a business is NOT heavily fined. Therefore, these two conditions are mutually exclusive. It is logically impossible for a business to both fail the inspection and receive a tax grant.

Step-by-Step Solution

1
Translate the premises into logical notation.
1. Fails Inspection -> Heavily Fined AND Mandatory Audit. 2. Mandatory Audit 'some' Tax Grant. 3. Tax Grant -> NOT Heavily Fined.
Formalizing the logic prevents confusion when combining statements and checking inferences.
2
Combine the first and third premises using the shared variable 'Heavily Fined'.
Fails Inspection -> Heavily Fined. The contrapositive of the third premise is: Heavily Fined -> NOT Tax Grant. Connecting them yields: Fails Inspection -> NOT Tax Grant.
Connecting conditionals reveals the implicit constraints that dictate what is logically impossible.
3
Evaluate the options against the deduced chain.
The option stating a business receives a grant and fails the inspection directly contradicts the deduction that failing the inspection guarantees NOT receiving a grant.
In a 'Cannot Be True' question, the correct answer must violate a stated premise or a valid deduction.

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Cannot Be True Questions
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