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Mentor Exercise

 

Directions: The following group of questions is based on a set of conditions. Choose the response that most accurately and completely answers each question. Hints, insights, partial solutions, and the answers are provided in the right-hand column.

Six people—Albert, Ben, Carrie, Darlene, Emily, and Fred—are competing in a gymnastics event. Two of them compete on the horse, two compete in the vault, and two compete on the parallel bars.
 
Ben competes on the horse if and only if Carrie competes in the vault.
 
If Darlene does not compete on the parallel bars, then Fred competes in the vault.
 
If Emily competes in the vault, then Fred does not.
 
This is a rather hard game. Its underlying structure is actually simple, but there’s lots of information to wade through. We start by symbolizing the conditions. We’ll use an equal sign to indicate that a person competes in a particular event. The first condition, “Ben competes on the horse if and only if Carrie competes in the vault,” can be symbolized as (B=H)<—>(C=V). The second condition, “If Darlene does not compete on the parallel bars, then Fred competes in the vault,” can be symbolized as (D≠P)—>(F=V). This in turn can be recast, using the contrapositive, as (F≠V)—>(D=P). Finally, the condition “If Emily competes in the vault, then Fred does not” can be symbolized as (E=V)—>(F≠V). This gives the following schematic:
 
 
(B=H)<—>(C=V)
 
(F≠V)—>(D=P)
 
(E=V)—>(F≠V)

To start the flow chart, look for the element that occurs in the greatest number of conditions; it is F. So build the chart around it. Start with the third condition:
 
(E=V)—>(F≠V)

 

Next, add the second condition:
(E=V)—>(F≠V)—>(D=P)

 

Finally, the condition (B=H)<—>(C=V) cannot be added to the chart, so it forms an independent flow chart:
(E=V)—>(F≠V)—>(D=P)
(B=H)<—>(C=V)
 
Note that A is “wild” since it is not contained in the diagram.
 
Question-1
If Ben competes on the horse, then which one of the following can be true?
  1. Both Emily and Albert compete in the vault.
  2. Emily competes on the horse and Darlene competes in the vault.
  3. Darlene does not compete on the parallel bars and Albert competes in the vault.
  4. Albert competes on the parallel bars and Carrie competes in the vault.
  5. Albert competes on the horse and Darlene does not compete on the parallel bars.
Solution
Since Ben competes on the horse, we know from the bottom half of the chart that Carrie competes in the vault. Furthermore, if Darlene does not compete on the parallel bars, then applying the contrapositive to the top part of the chart, we see that Fred also competes in the vault. This fills both slots for the vault, so no one else can compete in that event. These restrictions are sufficient to eliminate choices (A), (B), (C), and (E).
 
The answer is (D).
 
 
Question-2
If Darlene does not compete on the parallel bars, then which one of the following cannot be true?
  1. Ben competes on the horse.
  2. Fred competes in the vault.
  3. Albert competes on the parallel bars.
  4. Emily competes in the vault.
  5. Both Ben and Fred compete in the vault.
Solution
Apply the contrapositive to the top half of the diagram.
 
The answer is (D).
 
 
Question-3
If Ben and Carrie compete in the same event, then which one of the following can be true?
  1. Albert competes on the horse.
  2. Emily competes in the vault.
  3. Darlene does not compete on the parallel bars.
 
  1. I only
  2. II only
  3. III only
  4. I and III only
  5. I, II, and III
Solution
From the condition (B=H)<—>(C=V), we know that Ben and Carrie must both compete on the parallel bars. (Why?) As to I, since Albert is an independent element, we intuitively expect that he could compete on the horse, but you should verify this. As to II, if Emily competes in the vault, then from the diagram Darlene must compete on the parallel bars. This, however, puts three people in the parallel bar event, contradicting the condition that there are two people in each event. As to III, if Darlene does not compete on the parallel bars, then applying the contrapositive to the top diagram shows that Fred must compete in the vault, and Emily cannot compete in the vault. Now it’s easy to work out a schedule with these restrictions.
 
The answer is (D).
 
 
Question-4
Suppose the condition “If Carrie does not compete in the vault, then Emily does” is added to the given conditions. Which one of the following cannot be true if Emily and Darlene do not compete in the same event?
  1. Ben does not compete on the horse and Darlene does.
  2. Fred competes on the horse.
  3. Ben does not compete on the horse and Darlene competes on the parallel bars.
  4. Albert competes on the parallel bars.
  5. Emily competes in the vault.
Solution
This question is difficult because there are six different ways to assign different events to Carrie and Emily. Additionally, the string of inferences needed to answer the question is quite long.
 
To begin, add the new condition (C≠V)—>(E=V) to the diagram:
 
(C≠V)—>(E=V)—>(F≠V)—>(D=P)
 
(B=H)<—>(C=V)

Now assume that Ben does not compete on the horse and Darlene does, choice (A). Then use the above diagram to derive a contradiction—namely that Darlene also competes on the parallel bars.
 
The answer is (A).
 
 
Question-5
If Darlene competes in the vault, then how many different people could possibly compete on the horse?
  1. 2
  2. 3
  3. 4
  4. 5
  5. 6
Solution
Applying the contrapositive, along with the new condition “Darlene competes in the vault” to the original diagram, shows that Fred competes in the vault and Emily does not. Now Ben cannot compete on the horse. (Why?) From these conditions you should be able to work out three valid schedules.
 
The answer is (B).
 
 
Question-6
Suppose the condition “if Fred does not compete in the vault, then Emily does” is added to the original conditions. Of the following, which one cannot be true?
  1. Ben competes on the horse and Albert competes in the vault.
  2. Ben competes on the horse and Emily competes in the vault.
  3. Darlene competes on the parallel bars.
  4. Albert competes on the parallel bars.
  5. Fred competes on the parallel bars and Albert competes on the horse.
Solution
This question is hard, or at least long, because it actually contains five questions. The new condition changes the diagram only slightly:
 
(E=V)<—>(F≠V)—>(D=P)
 
(B=H)<—>(C=V)

Start with (A). If Ben competes on the horse, then from the bottom half of the new diagram Carrie must compete on the vault along with Albert. Turning to the top diagram, clearly Emily cannot compete in the vault, since that would put three people—Emily, Albert, and Carrie—in one event. But if Emily does not compete on the vault, then again from the top diagram and the contrapositive Fred must compete in the vault, which leads to the same contradiction.
 
The answer is (A).
 




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