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Difficulty: MediumForming Contrapositives

In an effort to promote sustainable industrial practices, the regional council has updated its environmental protection statutes. Under the new guidelines, a manufacturing facility is exempt from the annual carbon surcharge only if it relies primarily on renewable energy sources and does not discharge unmonitored effluent into adjacent waterways.

Which statement is logically equivalent to the principle governing the carbon surcharge exemption?

  1. If a manufacturing facility either does not rely primarily on renewable energy sources or discharges unmonitored effluent into adjacent waterways, then it is not exempt from the annual carbon surcharge.Answer
  2. B
    If a manufacturing facility relies primarily on renewable energy sources and does not discharge unmonitored effluent into adjacent waterways, then it is exempt from the annual carbon surcharge.
  3. C
    If a manufacturing facility does not rely primarily on renewable energy sources and it discharges unmonitored effluent into adjacent waterways, then it is not exempt from the annual carbon surcharge.
  4. D
    If a manufacturing facility is not exempt from the annual carbon surcharge, then it either does not rely primarily on renewable energy sources or discharges unmonitored effluent into adjacent waterways.
  5. E
    If a manufacturing facility either does not rely primarily on renewable energy sources or does not discharge unmonitored effluent into adjacent waterways, then it is not exempt from the annual carbon surcharge.

Answer

The statement asserting that if a manufacturing facility either does not rely primarily on renewable energy sources or discharges unmonitored effluent into adjacent waterways, then it is not exempt from the annual carbon surcharge.
The correct answer accurately forms the contrapositive of the principle in the stimulus. The stimulus establishes a conditional relationship where being exempt from the surcharge requires two necessary conditions: relying on renewable energy AND not discharging unmonitored effluent. To form the logically valid contrapositive, we must reverse the conditions and negate them. Negating a compound condition joined by 'and' requires applying De Morgan's laws, changing the 'and' to 'or' while negating both individual terms. Thus, the condition becomes 'does not rely on renewable energy OR discharges unmonitored effluent'. Placing this in the sufficient condition yields a statement that perfectly matches the correct option.

Step-by-Step Solution

1
Analyze the logical structure of the stimulus to identify the sufficient and necessary conditions.
Sufficient condition: Exempt from surcharge. Necessary condition: (Relies on renewable energy AND Does not discharge unmonitored effluent).
The phrase 'only if' introduces a necessary condition. Accurate baseline translation is required before manipulating the logic.
2
Begin forming the contrapositive by reversing the positions of the conditions and negating both sides.
NOT (Relies on renewable energy AND Does not discharge unmonitored effluent) -> NOT Exempt from surcharge.
A conditional statement is always logically equivalent to its contrapositive, which requires both reversal and negation.
3
Apply De Morgan's laws to simplify the negated compound necessary condition.
The negated condition becomes: '(Does NOT rely on renewable energy OR DOES discharge unmonitored effluent)'.
De Morgan's laws state that the negation of an 'AND' statement becomes an 'OR' statement where both individual terms are negated.
4
Combine the simplified terms to state the final, logically equivalent conditional statement.
If a facility does NOT rely on renewable energy OR it DOES discharge unmonitored effluent -> it is NOT exempt from the surcharge.
This completes the formation of the logically valid contrapositive, leading directly to the correct answer choice.

Key Concept

Forming Contrapositives of Compound Conditionals using De Morgan's Laws
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Topics

Forming Contrapositives

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