Forming Contrapositives

22 questions

All types22 Multiple choice12 True / False5 Matching5

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?

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.

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.

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.

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.

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.

Show answer & explanation

Answer: 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.

Answer

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.

Step-by-Step Solution

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.

Formulate the logical relationship.

Pass -> LIDAR AND (Decentralized OR Quantum)

This captures the compound nature of the necessary condition.

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'.

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.

Key Concept

Forming Complex Contrapositives with De Morgan's Laws

Question 22Question →

According to the newly ratified International Maritime Treaty, a sovereign vessel is permitted to navigate through the restricted ecological zone only if it possesses a valid ecological waiver or operates exclusively on electric power.

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

Any sovereign vessel that possesses a valid ecological waiver or operates exclusively on electric power is permitted to navigate through the restricted ecological zone.

Any sovereign vessel that neither possesses a valid ecological waiver nor operates exclusively on electric power is prohibited from navigating through the restricted ecological zone.

Any sovereign vessel that is prohibited from navigating through the restricted ecological zone must lack a valid ecological waiver and fail to operate exclusively on electric power.

Any sovereign vessel that lacks a valid ecological waiver or fails to operate exclusively on electric power is prohibited from navigating through the restricted ecological zone.

Any sovereign vessel that operates exclusively on electric power is automatically granted a valid ecological waiver.

Show answer & explanation

Answer: Any sovereign vessel that neither possesses a valid ecological waiver nor operates exclusively on electric power is prohibited from navigating through the restricted ecological zone.

Answer

The logically valid inference is that a vessel lacking both a valid ecological waiver and electric power cannot navigate the restricted zone.

The correct answer accurately applies De Morgan's laws to form a valid contrapositive. The stimulus dictates that permission to navigate requires at least one of two things: a waiver OR electric power. Therefore, if a vessel lacks BOTH (neither a waiver nor electric power), it has failed the necessary condition and must be prohibited from navigating.

Step-by-Step Solution

Diagram the original conditional statement from the stimulus.

Permitted to Navigate -> (Possesses Waiver OR Operates on Electric Power)

The phrase 'only if' introduces a necessary condition. Because there are two ways to meet it, we use the logical 'OR'.

Form the contrapositive by reversing the conditions and negating both sides.

NOT (Possesses Waiver OR Operates on Electric Power) -> NOT Permitted to Navigate

A valid logical inference from a conditional statement is its exact contrapositive.

Apply De Morgan's laws to simplify the negated compound necessary condition.

(NOT Possesses Waiver AND NOT Operates on Electric Power) -> NOT Permitted to Navigate

Negating an 'OR' statement turns it into an 'AND' statement. In plain English, 'not A or B' becomes 'neither A nor B'.

Match the resulting logical statement to the correct answer choice.

The correct option states that any vessel that neither possesses a waiver nor operates on electric power is prohibited from navigating.

This choice perfectly aligns with the simplified contrapositive diagram.

Key Concept

Forming compound contrapositives using De Morgan's laws (negating 'OR' into 'AND').

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Forming Contrapositives

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes