Grouping Setup Practice Drill #3

[mestifo]

Hi!

For the Grouping Practice Drill (3) on page 275, why does the book put rule one (If P serves on a subcommittee, then Q does not serve on that subcommittee) and rule 2 (R and S do not serve on the same subcommittee) in block? Arn't they if/then statements and treated like conditional logic statements? #HELP

[Dave Killoran]

Hi Mestifo,

Thanks for the question! Back on page 252, I talk about how blocks can be used in diagrams that have vertical components because grouping relationships presented in block form are more visually powerful. Since the groups in drill 3 have that vertical component, I went in that direction.

The nice thing is that the block representation is still conditional; it's just a different way of showing the relationship. Let's take a look at how that works:

• The rule in the drill states, "If P serves on a subcommittee, then Q does not serve on that subcommittee."
  
  The arrow diagram of that rule appears as: P Q
  
  We know that this rule (when considered with the contrapositive) means that when one of P or Q is on a subcommittee, the other is not on that subcommittee. That entire relationship can be represented as P Q.
  
  The P Q representation works just fine since it reflects the fact that the two can't be together on the same subcommittee. However, visually it doesn't match up well with the two subcommittees, which look like this:
  
  
  • __
    __ __
    __ __
    __ __
    3 4
  Well, they kind of look like that, at least However, if we use a vertical not-block to represent the PQ relationship, then it is easier to see that P and Q can't serve together on the same subcommittee. The meaning of the not-block is still that P and Q can't serve together, and since that is identical in meaning to the arrow diagram, we can use the representations interchangeably.

Ultimately, in certain instances blocks can be used to represent a conditional relationship (2 variables always together (such as in an "if and only if" relationship) and 2 variables never together (such as the one under discussion here) are the easiest to represent in this form) and every block you see actually reflects a conditional relationship of some sort.

Please let me know if that helps. Thanks!

[MRockwell]

Thank you for your reply. However, I had the same question and I am still confused. On page 266, you talk about "Negative Grouping Rule Misinterpretation". Under number 1, you explain that for the relationship A B: "While it is the case that A and B cannot be selected together, some students also mistakenly believe that it is impossible for A and B to both be absent." The possibilities resulting from this relationship are

1. A is selected, and B is not selected.
2. B is selected, and A is not selected.
3. Neither A nor B is selected.

Now for this A/B scenario, you are talking about the variables being selected or NOT being selected, whereas the question on page 275 is about whether a variable is being selected for one subcommittee or another. This must make a difference because in the solution to this question, you never say that though P Q results in P and Q being on different subcommittees, there is also the possibility of NEITHER of them being on the same subcommittee (which would mean BOTH are on the other subcomittee).So, can you explain what the difference is between these two questions? Why aren't there three possibilities for P/Q, just like the above example?

Furthermore, with this rule "If P serves on a subcommittee, then Q does not serve on that subcommittee," we know what happens when P serves, and we know what happens with Q doesn't serve because of the contrapositive, but we don't know what happens when P doesn't serve and when Q serves, right? (p. 269) So why is it true that P and Q MUST serve on different subcommittees if we don't know the result of these two sufficient variables?

Thank you!!

[Nikki Siclunov]

Dear MRockwell,

Let me jump in real quick and give you a few pointers.

Your understanding of the relationship between A and B is correct (A B). However, you are forgetting that the scenario in Drill #3 requires everyone to be in at least one of the two groups. So, whether you represent the first two rules using a double-not arrow or a Not-Block, the bottom line is this: all seven of the variables must be used between the two groups! Given that P and Q cannot serve on the same subcommittee, and also that neither of them can be left out, we have no choice but to reserve a spot for one of P or Q on each committee (hence, the P/Q designation in each group). The possibility you mention in your post (neither P nor Q being selected) would only be relevant if the scenario allowed for some of the senators to serve on neither committee, which was explicitly forbidden by the first rule.

You are correct in that a rule such as this (P Q) does not allow us to determine what happens to Q if P is not selected, nor does it tell us anything about P if Q is not selected. The only reason why P and Q must serve on different subcommittees is that every senator must serve on at least one.

Does that make sense? Let me know.

Thanks!

[MRockwell]

It DOES make sense! Lightbulb has been lit. Thank you for the reminder. I shall make sure I look out for the rule that everyone must be in one of the two groups. Thank you!

[yongjook]

Hi,

I have a quick question.

When I was diagrammed the first relation, I believe I made a mistake of trying to use the Two value system error on page 270-271.

I named the first group 1 and second ground 2 then stated P1 Q2 so the contrapositive would be Q1 P2 then i concluded using the two value system error that Q2 and P2 COULD be in the same group.

Now looking back I think I can only use the two value system error when the question actually states that Q is in group 1 or vice versa. Since the question simply states that p serves on A subcommittee, I shouldn't have done the two value system error.

Is this correct way of analyzing my mistake?

Thank you

[Nikki Siclunov]

Hi yongjook,

The issue you're facing has to do with the way you diagrammed the rule: your diagram only captures one aspect of the rule. Indeed, it is true that if P1 Q2 (and vice versa, if Q1 P2). If that were the full extent of the rule, then yes - P and Q could possibly end up in group 2 together without violating the conditional relationship between them.

However, as you note yourself, the rule states, "If P serves on a subcommittee...", i.e. on either subcommittee. Clearly, then, the same conditional relationship will exist if either P or Q served in group 2:

P2 Q1
Q2 P1

When you combine this with the diagram you wrote, we arrive at the following:

P1 Q2
Q1 P2

Or, more succinctly stated:

P Q

Hope this clears things up!

[srcline@noctrl.edu]

Good evening I have a clarification question on this drill. I understand the explanation that was given as to why V and T have to be on the same subcommittee, because the rule states that T serves on a subcommittee only if V serves on the same subcomittee. This is a conditional statement correct? If so I took the contrapostive of the rule and got if V does not serve on the same subcommittee then T does not serve on a subcommittee. So then I inferred that V and T have to be on the subcommittee to because they have to be on the same committee and if they werent there would be no room for both of them? That is why X is in Sub 1? Would looking at T and Vs contrapostive be helpful and can this be reliable when your trying to make inferences in grouping games?

Thankyou
Sarah

[Dave Killoran]

Hi Sarah,

Thanks for the questions on this one. If you have a chance, read my reply to your other question before reading this one.

Good evening I have a clarification question on this drill. I understand the explanation that was given as to why V and T have to be on the same subcommittee, because the rule states that T serves on a subcommittee only if V serves on the same subcomittee. This is a conditional statement correct?

Yes, it is a conditional rule. The "only if" is a necessary condition indicator, and that means that V is the necessary condition, leaving T to be the sufficient condition: T V.

If so I took the contrapostive of the rule and got if V does not serve on the same subcommittee then T does not serve on a subcommittee. So then I inferred that V and T have to be on the subcommittee to because they have to be on the same committee and if they werent there would be no room for both of them? That is why X is in Sub 1?

Yes, it's a space issue when you get down to placing T, V, and X. If we look at just those elements, it looks like this:

• Variables: T V X
  
  Rule: T V
  
  
  Available spaces:
  
  _____
  
  _____ _____
  1 2

So, if T goes in the first group, there's not enough room for V. So, T can't be in the first group, and must be in the second group. When T is placed in the second group, then V must go in the second group. That forces X into the first group:

• __V__
  
  __X__ __T__
  1 2

Ok, not the prettiest diagram, but I hope it gets the point across

Would looking at T and Vs contrapostive be helpful and can this be reliable when your trying to make inferences in grouping games?

Please see my other reply for my thoughts on how contrapositives work and how you should see them. Seeing contrapositives should be something that is second nature and nearly instant for you. They are always there, so you want to build towards becoming used to simply seeing them as part and parcel of the original statement.


Please let me know if that helps. Thanks!

[momgoingbacktoschool]

Hi Sarah,

Thanks for the questions on this one. If you have a chance, read my reply to your other question before reading this one.

Good evening I have a clarification question on this drill. I understand the explanation that was given as to why V and T have to be on the same subcommittee, because the rule states that T serves on a subcommittee only if V serves on the same subcomittee. This is a conditional statement correct?

Yes, it is a conditional rule. The "only if" is a necessary condition indicator, and that means that V is the necessary condition, leaving T to be the sufficient condition: T V.

If so I took the contrapostive of the rule and got if V does not serve on the same subcommittee then T does not serve on a subcommittee. So then I inferred that V and T have to be on the subcommittee to because they have to be on the same committee and if they werent there would be no room for both of them? That is why X is in Sub 1?

Yes, it's a space issue when you get down to placing T, V, and X. If we look at just those elements, it looks like this:

• Variables: T V X
  
  Rule: T V
  
  
  Available spaces:
  
  _____
  
  _____ _____
  1 2

So, if T goes in the first group, there's not enough room for V. So, T can't be in the first group, and must be in the second group. When T is placed in the second group, then V must go in the second group. That forces X into the first group:

• __V__
  
  __X__ __T__
  1 2

Ok, not the prettiest diagram, but I hope it gets the point across

Would looking at T and Vs contrapostive be helpful and can this be reliable when your trying to make inferences in grouping games?

Please see my other reply for my thoughts on how contrapositives work and how you should see them. Seeing contrapositives should be something that is second nature and nearly instant for you. They are always there, so you want to build towards becoming used to simply seeing them as part and parcel of the original statement.


Please let me know if that helps. Thanks!

Would it be incorrect to diagram T <--> V ?

Thank you!