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Zorluk: ZorForming Contrapositives

An autonomous vehicle's navigation system will pass the stringent Level 5 redundancy audit only if it incorporates dual-band LIDAR sensors, and operates with either a decentralized processing matrix or a quantum-encrypted communication link.

Which one of the following logically follows from the statement above?

  1. If an autonomous vehicle's navigation system does not incorporate dual-band LIDAR sensors, or if it operates with neither a decentralized processing matrix nor a quantum-encrypted communication link, then it will not pass the Level 5 redundancy audit.Cevap
  2. B
    Any autonomous vehicle's navigation system that incorporates dual-band LIDAR sensors, and operates with either a decentralized processing matrix or a quantum-encrypted communication link, will pass the Level 5 redundancy audit.
  3. C
    If an autonomous vehicle's navigation system does not pass the Level 5 redundancy audit, then it must either fail to incorporate dual-band LIDAR sensors, or it must operate with neither a decentralized processing matrix nor a quantum-encrypted communication link.
  4. D
    If an autonomous vehicle's navigation system lacks dual-band LIDAR sensors, or if it lacks a decentralized processing matrix or lacks a quantum-encrypted communication link, then it will not pass the Level 5 redundancy audit.
  5. E
    If an autonomous vehicle's navigation system lacks a quantum-encrypted communication link, it will not pass the Level 5 redundancy audit, even if it incorporates dual-band LIDAR sensors.

Cevap

The statement expressing that if the system does not incorporate dual-band LIDAR sensors, or if it operates with neither a decentralized processing matrix nor a quantum-encrypted communication link, then it will not pass.
The stimulus establishes a conditional rule: Passing the audit requires LIDAR, plus at least one of two other features (Decentralized processing OR Quantum link). The logically valid contrapositive dictates that if you fail to meet these minimum requirements, you fail the audit. Failing the requirements means either lacking the mandatory LIDAR, OR lacking both of the alternative features. The correct answer precisely maps this logic.

Adım Adım Çözüm

1
Identify the sufficient and necessary conditions in the stimulus.
Sufficient: Passes the Level 5 redundancy audit. Necessary: Incorporates dual-band LIDAR AND (operates with decentralized processing OR quantum-encrypted link).
The phrase 'only if' introduces the necessary conditions for passing the audit.
2
Formulate the logical relationship.
Pass -> LIDAR AND (Decentralized OR Quantum)
This captures the compound nature of the necessary condition.
3
Apply De Morgan's laws to find the contrapositive.
Not [LIDAR AND (Decentralized OR Quantum)] -> Not Pass. This becomes: (Not LIDAR) OR Not (Decentralized OR Quantum) -> Not Pass. Further simplifying: (Not LIDAR) OR (Not Decentralized AND Not Quantum) -> Not Pass.
A valid logical inference from a conditional statement is its exact contrapositive. Negating an 'AND' creates an 'OR', and negating an 'OR' creates an 'AND'.
4
Match the derived contrapositive to the answer choices.
The correct choice translates the mathematical contrapositive into plain English: missing LIDAR, or missing both Decentralized and Quantum, guarantees failing the audit.
The correct answer must perfectly mirror the valid contrapositive structure.

Anahtar Kavram

Forming Complex Contrapositives with De Morgan's Laws
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Konular

Forming Contrapositives

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