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Difficulty: Very hardChaining Conditional Statements

City planners have established new zoning rules for metropolitan districts. Under these rules, a district is zoned for mixed-use development only if both its transit hub is expanded and high-density housing is subsidized. The rules also stipulate that a district's commercial tax rate will increase if small business grants are not offered. Furthermore, high-density housing is subsidized only if the commercial tax rate does not increase, and unless the historic preservation board approves the plan, the transit hub will not be expanded.

Which one of the following conclusions can be logically drawn from the zoning rules described above?

  1. Any district that is zoned for mixed-use development must also offer small business grants.Answer
  2. B
    If a district offers small business grants and the historic preservation board approves its plan, the district will be zoned for mixed-use development.
  3. C
    If a district is not zoned for mixed-use development, then its commercial tax rate will increase.
  4. D
    Most districts that expand their transit hubs also offer small business grants.
  5. E
    Districts that offer small business grants are more likely to successfully attract mixed-use development.

Answer

Any district that is zoned for mixed-use development must also offer small business grants.
The stimulus provides a continuous logical chain. If a district is zoned for mixed-use development, it must subsidize high-density housing. Subsidizing high-density housing requires that the commercial tax rate does not increase. Because the tax rate will increase if small business grants are not offered, the fact that the tax rate does NOT increase means small business grants MUST be offered (the contrapositive of the second rule). Linking these together demonstrates that zoning for mixed-use development logically guarantees that small business grants are offered.

Step-by-Step Solution

1
Translate the first zoning rule into formal logic.
Mixed-use development (MUD) -> Transit hub expanded (TE) AND High-density housing subsidized (HDS).
The phrase 'only if both' indicates that expanding the transit hub and subsidizing high-density housing are necessary conditions for mixed-use development.
2
Translate the second zoning rule into formal logic.
Small business grants NOT offered (~SBG) -> Commercial tax rate increases (CTR).
The word 'if' introduces a sufficient condition. A lack of grants guarantees a tax rate increase.
3
Translate the third zoning rule into formal logic.
High-density housing subsidized (HDS) -> Commercial tax rate does NOT increase (~CTR).
The phrase 'only if' introduces a necessary condition. Subsidizing housing requires that the tax rate remains stable.
4
Form the contrapositive of the second rule to establish a logical bridge.
Commercial tax rate does NOT increase (~CTR) -> Small business grants offered (SBG).
To connect the chain from ~CTR, we must negate both sides of the second rule (~SBG -> CTR) and reverse them.
5
Chain the conditional statements together starting from mixed-use development.
MUD -> HDS -> ~CTR -> SBG.
Following the logical links from the sufficient condition (MUD) leads directly to the necessity of offering small business grants (SBG).

Key Concept

Chaining Conditional Statements
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Chaining Conditional Statements

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