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Difficulty: MediumDiagramming Conditional Statements

Match each of the following culinary certification rules to its correct logical contrapositive diagram.

Assume the following variables:
MM = Master status is achieved
TT = Technique is perfected
FF = Final exam is passed
EE = Critical error occurs

  • Master status is achieved only if the technique is perfected.∼T→∼M\sim T \rightarrow \sim M
  • Failing to perfect the technique ensures that the final exam is failed.F→TF \rightarrow T
  • The final exam is passed unless a critical error occurs.∼F→E\sim F \rightarrow E
  • No master status is achieved without passing the final exam.∼F→∼M\sim F \rightarrow \sim M

Answer

The correct matches require translating each English conditional statement into its baseline logical diagram, and then finding the logically equivalent contrapositive for each among the choices.
Each correct match pairs the original English statement with its logically equivalent contrapositive. By successfully identifying the sufficient and necessary conditions encoded in terms like 'only if', 'ensures', 'unless', and 'no... without', you derive the correct baseline diagram. Negating and reversing both terms then yields the provided answers.

Step-by-Step Solution

1
Translate rule 1 and find its contrapositive.
'Master status is achieved only if the technique is perfected' translates to M→TM \rightarrow T. The contrapositive is ∼T→∼M\sim T \rightarrow \sim M.
The phrase 'only if' always precedes the necessary condition.
2
Translate rule 2 and find its contrapositive.
'Failing to perfect the technique ensures that the final exam is failed' translates to ∼T→∼F\sim T \rightarrow \sim F. The contrapositive is F→TF \rightarrow T.
The word 'ensures' functions as a sufficient condition indicator.
3
Translate rule 3 and find its contrapositive.
'The final exam is passed unless a critical error occurs' translates to ∼E→F\sim E \rightarrow F. The contrapositive is ∼F→E\sim F \rightarrow E.
The word 'unless' is logically equivalent to 'if not', negating the other term to form the sufficient condition.
4
Translate rule 4 and find its contrapositive.
'No master status is achieved without passing the final exam' translates to M→FM \rightarrow F. The contrapositive is ∼F→∼M\sim F \rightarrow \sim M.
The formula 'No X without Y' means that Y is required for X, thus X is the sufficient condition and Y is the necessary condition.

Key Concept

Translating diverse conditional language into logical symbols and recognizing contrapositives.
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Topics

Diagramming Conditional Statements

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