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Difficulty: HardMust Be True / Most Strongly Supported

The regional groundwater protection board established the following regulatory criteria for industrial operations:

• Every industrial facility operating within the Karst Protection Overlay District that discharges effluent into the primary watershed is required to maintain a tertiary closed-loop filtration system.
• Any industrial facility that maintains a tertiary closed-loop filtration system is exempt from quarterly bioassay soil monitoring.
• No industrial facility subject to mandatory annual hydrologic reporting is exempt from quarterly bioassay soil monitoring.
• Some manufacturing plants operating within the Karst Protection Overlay District are subject to mandatory annual hydrologic reporting.

If all of the statements above are true, which of the following statements must also be true? (Select all that apply.)

  1. At least some manufacturing plants operating within the Karst Protection Overlay District do not discharge effluent into the primary watershed.Answer
  2. B
    Every industrial facility operating within the Karst Protection Overlay District that is exempt from quarterly bioassay soil monitoring discharges effluent into the primary watershed.
  3. C
    All industrial facilities that do not discharge effluent into the primary watershed are subject to mandatory annual hydrologic reporting.
  4. No industrial facility subject to mandatory annual hydrologic reporting maintains a tertiary closed-loop filtration system.Answer
  5. E
    Any manufacturing plant operating within the Karst Protection Overlay District that does not maintain a tertiary closed-loop filtration system violates board regulations.

Answer

The statements establishing that at least some manufacturing plants in the Karst District do not discharge effluent into the primary watershed, and that no facility subject to mandatory hydrologic reporting maintains a tertiary filtration system, must both be true.
The deduction that no facility subject to mandatory reporting maintains tertiary filtration strictly follows from chaining the rule that reporting facilities are never exempt from bioassay monitoring with the contrapositive of the filtration-exemption rule. Furthermore, because some Karst plants are subject to mandatory reporting, those specific plants cannot possess tertiary filtration, which in turn proves via contrapositive that they cannot be discharging effluent into the primary watershed.

Step-by-Step Solution

1
Symbolize the conditional premises and universal rules.
Let K = Karst District, D = discharges effluent, T = tertiary closed-loop filtration, E = exempt from quarterly bioassay monitoring, R = subject to mandatory annual hydrologic reporting. Premises: (1) (K and D) -> T; (2) T -> E; (3) R -> not E; (4) Some manufacturing plants (M) satisfy (K and R).
Formalizing the relations establishes the exact deductive pathways and contrapositives.
2
Chain the conditional statements linking hydrologic reporting to tertiary filtration.
From T -> E, the contrapositive is not E -> not T. Combining with R -> not E yields R -> not E -> not T. Thus, R -> not T (no reporting facility maintains tertiary filtration).
This verifies the universal deduction regarding reporting facilities.
3
Combine the existential quantifier with the conditional constraints in the Karst District.
There exists a plant with (M and K and R). Since R -> not T, this plant has not T. From (K and D) -> T, the contrapositive is not T -> (not K or not D). Because this plant is in the Karst District (K is true), not D must be true. Therefore, this plant does not discharge effluent into the primary watershed.
This establishes that at least some Karst manufacturing plants do not discharge effluent.

Key Concept

Quantifier Intersection and Conditional Contrapositive Chaining
Estimated Time:2m 0s
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Topics

Must Be True / Most Strongly Supported

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