Setup and Rule Diagrams

[Administrator]

Setup and Rule Diagram Explanation

This is a Pure Sequencing game.

The game scenario establishes that seven librarians are scheduled for desk duty for one week—Monday through Saturday. Except for Saturday when two librarians must be on duty, there are no ties. The following linear scenario underpins the sequence:

June15_game_3_diagram_1.png

Although Pure Sequencing games involve relationships that are relative and not precisely fixed, a linear diagram can help us represent inferences that could result from the application of the rules.

The first rule establishes the following sequence:

• H L

The second rule establishes the following sequence:

June15_game_3_diagram_3.png

The third rule establishes the following sequence:

June15_game_3_diagram_4.png

The fourth rule establishes the following sequence:

• K Z

When combined, the first four rules produce the following sequencing chain:

June15_game_3_diagram_6.png

Since only F and H could be scheduled for duty on Monday, we can represent this as a Dual Option on our Linear setup. Also, since two librarians must be scheduled for duty on Saturday, it follows that two of G, L, and Z must be scheduled on Saturday:

June15_game_3_diagram_7.png

The fifth rule establishes the following conditional sequence between F and L:

June15_game_3_diagram_8.png

The implications of the contrapositive deserve a closer look. If L is not on duty on Saturday, then the two librarians on Saturday must be G and Z:

June15_game_3_diagram_9.png

Since L must be scheduled earlier than F, and H must be earlier than L (first rule), the placement of H, L and F is now completely determined:

June15_game_3_diagram_10.png

Thus, our final diagram looks like this:

June15_game_3_diagram_11.png

[Cowboys1118]

Why is K not included as possibly being scheduled for Saturday?

Thanks!

[Adam Tyson]

Because K is always before Z, Cowboys! If K was on Saturday, it wouldn't be before anything, but would be tied for "last", violating that sequencing rule. The only variables that can be on Saturday are the ones that don't have to be before anything else.

I hope that helps clear thing up!

[az305203]

For the conditional relationship, I saw it as mutually exclusive and diagrammed it as such:

F L LS
But obviously if LS then F would have to be before L since Saturday is the last day so it is essentially a Double Arrow relationship

And LS L F
But obviously if L is before F then it can't be on the last day so that is also essentially a Double Arrow relationship

I did everything else with the diagram correctly per the explanation (including the diagram for the implications of the contrapositive), but because of the nature of the conditional relationship I just diagrammed the rule as
L F
OR
LS

Is there something that I was overlooking in my analysis of that, or is that functionally correct in working with the conditional?

Thank you!

[Adam Tyson]

That approach works in this case, az305203! While most conditional rules allow for three possibilities - the original claim, the contrapositive, and the often overlooked situation where the sufficient condition does not occur and the necessary condition occurs anyway - in this instance that third possibility is out. There is no way for L to be on Saturday AND for F to not be before L. They can't tie, and there is no place to go that is after Saturday. So, it IS a double arrow relationship! Either F is before L and L is on Saturday, or F is NOT before L (and is therefore after it) and L is not Saturday. Both conditions occur, or neither does. Perfect!

[Mariam]

hello- I am a little confused about the sequence when L is not on Saturday. I only had that if L is not on Saturday L-F and H has to be on Monday, L on Tuesday and F on Wednesday with K-Z and M-G on Thursday to Saturday. I don't understand why G and Z have to be on Saturday. Why can't the sequence be HLFKZ and MG on Saturday or HLFMG and KZ on Saturday?
Thanks!

[Adam Tyson]

M cannot be on Saturday because it must be sometime before G, Mariam, and K cannot be on Saturday because it must be sometime before z. The only variables eligible to be on Saturday are L, G and Z, because they are the only variables that don't have to have something after them, so once one of them is NOT on Saturday, the other two have to be!

[Insiya]

Hi,

I'm having trouble understanding the last rule of this game and how the contrapositive of this rule forces G and Z into the spots on Saturday. Can you please explain why the contrapositive of this rule needs to be taken and why the regular rule cannot be applied where there is the possibility that L is on Saturday?

Thanks!

[Rachael Wilkenfeld]

Hi Insiya,

There's nothing about the last rule in particular that forces G and Z on Saturday. It's just the operation of the rest of the sequence.

In order for a librarian to be on Saturday, on one can be before that variable.

F is earlier than K and M. F cannot be on Saturday.
G is not earlier than anyone. G can be on Saturday.
H is earlier than L. H cannot be on Saturday.
K is earlier than Z. K cannot be on Saturday.
Unless L is on Saturday, L must be earlier than F. L can be on Saturday.
M is earlier than G. M cannot be on Saturday.
Z is not earlier than anyone. Z can be on Saturday.

That only leaves three options for the two slots on Saturday: G, L, and Z. Once one of them is not on Saturday, the other two must be in order to fill it up.

The last rule and contrapositive are logically identical, so whichever is easiest for you to understand is fine to write out.

Hope that helps!

[g_lawyered]

Hi P.S.,
I have a question about implication of rule 5 (the statement and the contrapositive). I understand why the contrapositive of L NOT being in Saturday leads to the possibility in the explanation above. HOWEVER, my question is, can we make an inference on the conditional statement (without committing a mistaken reversal- otherwise known as going against the arrow)? Here's what I mean:
F-- L L on Sat

Because of the Not-Laws from the other rules, can we infer that when L is on Sat, L must go in with either G or Z? This would fill up the options Sat. has. I know that the inference doesn't lead to more established places the rest of the variables are placed in, but is this a valid inference?

Thanks in advance!