Hey cornhusker,
Thanks for the questions! This game is notoriously tricky, and the contrapositives in particular seem to give people trouble, so let me see if I can break these down for you and show you the proper way to approach it.
For the third rule, you are correct in recognizing that it essentially produces two separate ideas:
(1) Not Y, then either L or O:
Y 
L or O
(2) Not Y, then NOT L or NOT O:
Y L or
O (this is
the same as saying they cannot both be in the park)
Your double-not arrow diagrams are technically correct as well, although I think in this case probably unnecessary. To me they make it more confusing than simply showing the two diagrams as I've done above.
Note: double-not arrows are really effective at showing single relationships, like two variables that cannot both be selected (A
B), or two variables that cannot both be absent (Not A
Not B). But using them as part of a larger diagram with multiple sufficient or necessary conditions is likely to be more trouble than it's worth.
This is where you ran into trouble. If Y is not in the park, only
two possibilities exist:
(1) L is in the park, and O is not
(2) O is in the park, and L is not
Your point 4.c) below indicates the possibility that neither L nor O is in the park when Y is absent, and this is a violation of the rule: "either L or O...are in the park." They cannot BOTH be there, but one of them must be if Y is not!
With that in mind, let's consider the contrapositives of both statements I gave showing the two separate ideas in rule #3:
(1)
Y L or O ; contrapositive:
L and
O Y
(2)
Y L or
O ; contrapositive: L and O
Y
Essentially (1) tells us that if both L and O are missing, Y must be in the park (Y's absence would force one of L or O in). And (2) tells us that if both L and O are in the park, Y must be as well (Y's absence would mean they cannot both be there).
And that's it! In summation, here are the possible scenarios for Y, L, and O:
(1) Y out, so L is in and O is out
(2) Y out, so O is in and L is out
(3) L and O both in, so Y is in (guaranteed if L and O are known to be in)
(4) L and O both out, so Y is in (again, known if L and O are both out)
(5) Y in, and then L and O are totally free to be in or out
I should point out that I wouldn't go to anywhere near this degree of analysis on test day—just show rules and, if needed, contrapositives—but hopefully it gives you a better sense of exactly what that rule itself tells us.
For your next question concerning rule #4 (a trickier rule, in my opinion), we once again have a two-part statement: if either L or O is not in the park, then F and S are both in. The "not L" or "not O" sufficient conditions would allow us to represent this with two diagrams:
(1)
L F and S ; contra:
F or
S L
(2)
O F and S ; contra:
F or
S O
Do you see how these would also cover the condition where
both L and O are not in? In that case it would still be clear that F and S were in the park. Just remember that multiple conditions with "or" or with "and" require that you change between the two as you take contrapositives. So "or" becomes "and" and the terms get negated, and vice versa.
Two things in conclusion. First, the two diagrams with F and S above capture the fourth rule well, but it is also possible, if you choose, to combine them into a single diagram with a single contrapositive:
L or
O F and S ;
contra:
F or
S L and O
Just keep in mind that the "or" here means "if either type is gone," and not that "they must both be gone."
Secondly, the third and fourth rules can be connected to make further inferences, based on the shared variables L and O. For instance, consider what we could deduce if the park does not contain Y:
Y L or
O F and S
And from the second rule:
F
P , so
Y P
Meaning, from the contrapositive of that:
P
Y
And so on! Tricky game, but a fantastic exercise for practicing with diagramming multiple-condition statements and their contrapositives, as well as for making inferences with them.
Give it another look and let me know if this helps!
Jon
