Setup and Rule Diagrams

[Dave Killoran]

Setup and Rule Diagram Explanation

This is an Undefined Grouping Game.

The game is Undefined because we do not know how many photographs are displayed in the album, nor do we know exactly how many people are in a photograph. With these elements unspecified, we cannot create a diagram in the traditional sense; that is, we cannot create a group of say, five spaces and then place variables in those spaces. Instead, we must diagram the rules and make inferences, and then proceed to the questions without a defined group in place.

The Rules

Rule #1. One of the easiest mistakes to make in this game is to misinterpret the wording in each of the rules. This rule indicates that whenever S appears in a photograph, then W appears in that same photograph. The correct diagram for that relationship is:


S W


Many students mistakenly reverse the above diagram. To avoid doing so, consider the rule for a moment: does the rule say that every time W appears in a photograph that S also appears? No. Although the difference is subtle, the wording in the rule is that W appears in every photograph that S appears in. Thus, S is the sufficient condition, and the appearance of S indicates that W will appear in that same photograph.

Rule #2. The wording in this rule is identical to the wording in rule #1, and so it is just as easy to make an error when diagramming this rule. The correct diagram is:


U S


When we get to the “Inference” section we will discuss how the first and second rules can be linked. Also, note that since both of the first two rules are “positive” (no negative terms involved) we are not diagramming contrapositives since you should know those as an automatic result of seeing any conditional statement.

Rule #3. This is also a tricky rule, not only because of the wording, but also because the conditional relationship between R and Y is easy to misunderstand. First, let us diagram the rule:


Y R


Since this rule contains a negative, let us also diagram the contrapositive:


R Y


Many students will interpret these two diagrams as reduced to a simple double-not arrow relationship: Y R. This is not the meaning of the rule! Instead, consider the exact relationship between Y and R: when Y is not in the photograph, then R must be in the photograph; and, via the contrapositive, when R is not in the photograph, then Y must be in the photograph. Thus, when one of the two is not in the photograph, the other must be in the photograph, and that relationship can best be expressed by stating that both cannot be absent from the photograph. The correct double-not arrow diagram, then, is:


R Y


The operating result of this rule is that either Y or R, or both, must appear in each photograph (note that R and Y can appear in a photograph together; the rule does not prohibit this occurrence).

Rule #4. This rule is actually two rules in one: one rule states that T and W do not appear in the same photograph, and the other states that R and W do not appear in the same photograph. Let’s consider each part separately.

As stated directly, when W is in a photograph, T is not in that same photograph:


W T


This rule can be turned into the a double-not arrow indicating that T and W are never in the same photograph:


W T


The other part of the rule states that when W is in a photograph, R is not in that same photograph:


W R


This rule can be turned into the a double-not arrow indicating that R and W are never in the same photograph:


W R


One note about the rules: every variable is mentioned in the rules except Z. Thus, Z is a random in this game, and since there is not a specified number of spaces, Z is largely powerless in this game. If a question stem includes Z as part of a local condition, there must be other information included in the question stem, and you should focus on the other information first.

In review, the four rules contain five basic grouping relationships that create many different inferences. When you consider the four rules, you can see that the test makers placed a trap for the unwary student. A student who quickly reads through the rules and does not read for meaning can easily mis-diagram one or more of the rules, and of course mis-diagramming during the setup is almost always costly.

After correctly negotiating each of the rule diagrams, the next step is to make inferences by connecting the rules. In this game, there are many inferences, and so the challenge becomes managing the information.


The Inferences

Inference #1. By connecting the first and second rules, we can create the following chain:


U S W


This connection is important because it shows that if U appears in a photograph then two other friends must also appear in that same photograph. Via the contrapositive, the relationship also indicates that if W does not appear in a photograph, then neither S nor U can appear in that photograph.

Inferences #2 and #3. The first and fourth rules can be connected through W:


S W T

+

S W R



These two relationships yield the following inferences:


S T

+

S R


Inferences #4 and #5. The two previous inferences connected S to T and R through S’s relationship with W. Since U also has a relationship with S from the second rule, we can connect the second rule to inferences we just made:


U S T

+

U S R


These two relationships yield the following inferences:


U T

+

U R


Inference #6. Because R appears in the third and fourth rules, we can make a connection using R:


Y R W


This relationship results in the unique inference that:


Y W



Because both conditions are negative, take the contrapositive:


W Y


Thus, if W appears in a photograph, then Y must appear in that photograph as well.


Inferences #7 and #8. From the inference we just made we know that when W is in a photograph, then Y must also be in that photograph. We can add this to the chain that appeared in inference #1 (which was the combination of the first two rules):


U S W Y


These relationships yield the following two inferences:


S Y

+

U Y


Thus, the appearance of either S or U will ultimately force Y to appear. This makes U a powerful variable: when U appears in a photograph, then S, W, and Y must also appear, and R and T cannot appear. Thus, the appearance of U allows for only two solutions, depending on whether Z is in the photograph.


There are other ways to arrive at some of these inferences (for example, when S has a relationship with a variable, U has the same relationship because of the second rule), but each inference above is a product of combining the rules or of combining the rules and inferences (do not forget to recycle your inferences!).

Compiling all of the information above, we arrive at the final setup for this game:

pt45_d04_g3_20a.png

pt45_d04_g3_20b.png

As long as you focus on connecting the variables in the rules and inferences, you can attack the questions and complete each problem with relative ease. Remember, this game is not logically difficult; it is simply an information management game, so do not be intimidated!

[seraphinali]

In this game (Page235),
Rule 1: not Y →R,
Rule 4: double-not arraow between W and R ,
then reference#6: not Y → R→ not W;

I can understand such inference, but my question is: if we are using the diagram, it seems that we could more easily infer a double-not inference between not Y and W, is something wrong with this? If yes, why?

Thanks!

[Dave Killoran]

Hey Seraphinali.

Thanks for the question--it's a good one. You certainly can draw that inference, so there's no problem there. But, there are a few reasons I prefer the W → Y representation instead:

1. They are different representations of the same idea, but W → Y is far more direct than not Y ←|→ W.

W → Y tells you that when W is the photograph, Y must also be in the photograph. Thus, you can key on the presence of W (or, via the contrapositive, on the absence of Y) while solving questions.

Using not Y ←|→ W, when you have W, you actually cannot have not Y; in other words, you must have Y. But reaching that conclusion requires you to process a double not, which, while easily doable, still requires an extra processing step (the same thing occurs when no Y occurs). So, to me, W → Y is actually more efficient because it takes fewer steps to process.

2. For most people, not Y ←|→ W is harder to use.

not Y ←|→ W actually means the same thing as W → Y, namely that you can't have W and not Y occur simultaneously (because when W is present then Y is present). But, for most people, using that representation doesn't seem as clear, and it is easier to get confused (which is obviously deadly during a game).


There's an element of personal preference in this: if you prefer using not Y ←|→ W and you are comfortable with it, then by all means do so. Functionally it represents the same idea as W → Y, so you will not making a logical error or harming your testing in any way. I personally think W → Y is easier to use (for me, and I think most people) but the law of test taking is that you must use the tools that feel the most comfortable and logical to you.

Please let me know if that helps. Thanks!

[seraphinali]

Thank you, Dave!
I agree that using the double-not arrow costed more time to get to W→Y so as to find the correct answer. That's why I thought maybe something wrong there.

Thanks again for the explanation.

Seraphina

[Fish]

Hi,
How do I know which way the letters are supposed to go. For example I keep putting
R----->Y when according to the book it should be the other way around.

Also can I assume that the statements below are true:

Y---> R
R----> Y
then both can be in the group together R<----/---->Y

and

Y----> R
R---->Y
then only one can be in a group at a time R<-----/----->Y

with Y or R meaning that the letter would be crossed out.

Thanks!

[Nikki Siclunov]

The rule states:

R appears in every photograph that Y does NOT appear in.

So, if you look at a photograph and you don't see Y in there, you must see R:

no Y ---> R

By the contrapositive, imagine you see a photograph in which there is no R. Clearly, Y cannot be missing: if it were, then R would have to be there instead. So, Y must appear in that photograph:

no R ---> Y

Taken together, the two statements mean that you cannot have a photograph in which both R and Y are missing: as long as one of them is missing, the other one has to be there. In other words, at least one of R or Y must be in the photograph at all times. However, this does not rule out the possibility that R and Y are both present (i.e. you don't have a rule stating if R ----> no Y).

If you prefer to use the double-not arrow, in this case it would look like this:

no R <---/---> no Y

(i.e. if you don't have R, then you cannot NOT have Y, meaning you need to have Y; and vice versa).

However, I would not use the double-not arrow in a rule with a negative sufficient condition unless you are really comfortable with it; otherwise is it could confuse you.

Good luck!

[Patrick.a.anderson]

Question #17: I am unsure why answer choice C could be true. The solution says that WSY is possible, but I don't understand how. Because S appears in a photo, I would think U should appear in the same photo as well.

U -> S

If no S, then no U.

Therefore U cannot be alone, and U must be with S, which must be with W.

I my diagram, I have a WSUY block.

[Adam Tyson]

Patrick,

It looks like you have fallen into a fairly common trap of conditional reasoning. Let's take a look at the rule that you've focused on here:

U -> S (if U is present, S must also be present)

You correctly identified the contrapositive, S ->U

In the original rule, U is SUFFICIENT for S; S is NECESSARY in the presence of U. Remember, though, that the Necessary condition can happen whether the Sufficient condition occurs or not. If U is present, S must be, but S can also be in a photo without U. To assume that whenever S is present, U must also be present is to make a mistaken reversal of the original rule. That would look like this: S -> U

Take another look at the lesson on conditional reasoning, and beware of mistaken reversals and mistaken negations. Mastering this kind of reasoning will take your game to a whole new level.

Hope that helped! Good luck,

Adam

[salty]

Hi,

On the Logic Games Bible book pg.309, the question about 7 friends in an album, the 4th rule "Neither Ty nor Raimundo appears in any photograph that Wendy appears in", its explanation on the next page says that it is a two rules in one and separates the two conditions. But I was wondering just as "If A or B is selected, then C must be selected" can be diagrammed as "A or B C", if the 4th rule could be diagrammed as "W T-slash (as in negative) or R-slash(as in negative)". If not, I would like to know why not.


I thank you in advance for your time and help.

[Dave Killoran]

Hi Salty,

Yes, the fourth rule can be diagrammed as one consolidated statement. That diagram would be a little different from what you stated however, It would be:

T
W and
R

(the difference is "and" instead of "or")

The reason I diagrammed it as two separate statements there is because each individual statement is an extremely powerful negative grouping relationship.

Please let me know if that helps. Thanks!