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Question

Difficulty: EasyForming Contrapositives

The university's graduate committee has recently updated the financial aid regulations for all incoming and returning students. The new guidelines establish strict conditions for the most prestigious awards. According to the official policy, being awarded the Chancellor's Fellowship requires that a student receive a full tuition waiver and be granted a monthly research stipend.

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

  1. A
    If a student receives a full tuition waiver and is granted a monthly research stipend, then that student is awarded the Chancellor's Fellowship.
  2. B
    If a student is not awarded the Chancellor's Fellowship, then that student does not receive a full tuition waiver and is not granted a monthly research stipend.
  3. If a student does not receive a full tuition waiver or is not granted a monthly research stipend, then that student is not awarded the Chancellor's Fellowship.Answer
  4. D
    If a student receives a full tuition waiver or is not granted a monthly research stipend, then that student is not awarded the Chancellor's Fellowship.
  5. E
    If a student is not awarded the Chancellor's Fellowship, then that student does not receive a full tuition waiver or is not granted a monthly research stipend.

Answer

If a student does not receive a full tuition waiver or is not granted a monthly research stipend, then that student is not awarded the Chancellor's Fellowship.
The correct answer accurately forms the contrapositive of the compound conditional statement in the stimulus. The original statement establishes that the Chancellor's Fellowship is sufficient to guarantee both a full tuition waiver and a monthly research stipend (Fellowship -> Waiver AND Stipend). To form the logically valid contrapositive, we must negate the necessary conditions and reverse the direction of the arrow. According to De Morgan's laws, negating an 'AND' statement creates an 'OR' statement. Therefore, lacking either a full tuition waiver or a monthly research stipend is sufficient to conclude that the student was not awarded the Chancellor's Fellowship.

Step-by-Step Solution

1
Identify the formal conditional relationship presented in the stimulus.
The stimulus uses the word 'requires' to introduce necessary conditions. The relationship is: Chancellor's Fellowship -> Full tuition waiver AND Monthly research stipend.
Extracting the formal logical structure from the text is the necessary first step to determine what logically follows.
2
Apply the rules of formal logic to construct the contrapositive of the statement.
To form the contrapositive, reverse the terms and negate them. The negation of 'Waiver AND Stipend' requires applying De Morgan's laws, becoming 'Not Waiver OR Not Stipend'.
The contrapositive is the only conditional statement that is logically equivalent to the original statement.
3
Combine the reversed and negated terms to form the final deduction.
The valid deduction is: Not Waiver OR Not Stipend -> Not Chancellor's Fellowship.
This guarantees that if either necessary condition is missing, the sufficient condition cannot have been met.

Key Concept

Forming the contrapositive of a compound conditional statement using De Morgan's laws.
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Topics

Forming Contrapositives

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