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Difficulty: HardDiagramming Conditional Statements

According to the newly ratified Urban Development Code, a commercial plaza will be granted a high-density zoning designation only if it both includes a subterranean transit link and incorporates advanced seismic base isolators. Furthermore, no plaza can incorporate advanced seismic base isolators unless it utilizes aerospace-grade titanium framing. Finally, any plaza that includes a subterranean transit link is automatically exempt from standard municipal parking minimums.

Which one of the following statements can be logically deduced from the regulations above?

  1. If a commercial plaza does not utilize aerospace-grade titanium framing, it will not be granted a high-density zoning designation.Answer
  2. B
    Any commercial plaza that utilizes aerospace-grade titanium framing and is exempt from standard municipal parking minimums must be granted a high-density zoning designation.
  3. C
    If a commercial plaza is not granted a high-density zoning designation, then it does not utilize aerospace-grade titanium framing.
  4. D
    If a commercial plaza does not include a subterranean transit link, it can still be granted a high-density zoning designation provided that it utilizes aerospace-grade titanium framing.
  5. E
    Most commercial plazas that are exempt from standard municipal parking minimums also incorporate advanced seismic base isolators.

Answer

If a commercial plaza does not utilize aerospace-grade titanium framing, it will not be granted a high-density zoning designation.
The correct answer accurately applies the contrapositive to the logical chain established in the stimulus. The rules dictate that High-Density Designation requires Advanced Seismic Base Isolators, which in turn require Aerospace-Grade Titanium Framing (HDD -> ASBI -> ATF). By taking the contrapositive of this entire chain, we can deduce that if a plaza lacks Titanium Framing (Not ATF), it cannot have Base Isolators (Not ASBI), and therefore cannot receive the High-Density Designation (Not HDD).

Step-by-Step Solution

1
Diagram the first compound conditional statement.
High-Density Designation (HDD) -> Subterranean Transit Link (STL) AND Advanced Seismic Base Isolators (ASBI)
The phrase 'only if' introduces a necessary condition. Because there is an 'and', both conditions are necessary for the high-density designation.
2
Diagram the second conditional statement.
Advanced Seismic Base Isolators (ASBI) -> Aerospace-Grade Titanium Framing (ATF)
The phrase 'no [X] unless [Y]' translates logically to 'If X, then Y'.
3
Diagram the third conditional statement.
Subterranean Transit Link (STL) -> Exempt from Parking Minimums (EPM)
The word 'any' introduces a sufficient condition.
4
Chain the statements together and form the contrapositives to find the correct deduction.
Chain 1: HDD -> ASBI -> ATF. Contrapositive: Not ATF -> Not ASBI -> Not HDD.
Chain 2: HDD -> STL -> EPM. Contrapositive: Not EPM -> Not STL -> Not HDD.
Valid deductions must follow the direction of the conditional arrows or rely on their contrapositives.

Key Concept

Forming deductive chains and contrapositives from compound conditional statements
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Topics

Diagramming Conditional Statements

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