bp shinners wrote:PLXTDNR wrote:Hey there bp - I'm hoping you can explain a rule on games to me -

the dreaded "if and only if."

1) A goes to the party if, and only if, B goes to the party AND

2) If A goes to the party then B goes to the party.

Now, the way it looks to me is in scenario 1, they must go together. A goes to the party if B goes to the party, and does not go to the party without B.

In Scenario 2, it looks like B is free to go to the party without A, but A must have B. (If A then B, if no B then no A)

I think I have that right (but am not sure), but what would the contrapositive for scenario 1 look like?

Thanks!

You've got it exactly right.

If and Only If statements can also be called Must Be Together statements - you either have both, or you don't have either. As such, we write it out as a biconditional:

A<-->B

That way, I know it goes both ways - If A, then B; If B, then A.

The contrapositive (if you want to call it that) is:

NOT A<--> NOT B

If no A, then no B; if no B, then no A

For grouping games, these are unbelievably powerful rules, and they usually lead to scenarios.

The second rule you state is what we call a Tag Along rule, and they're the weakest of the four main types of grouping game rules.

For the record, those rules are:

AT LEAST ONE

-The sufficient condition is negated

-So NOT A-->B is the same as saying you must have A or B (or both, since or is inclusive)

-Shortcut: If the sufficient condition is negated, you must have at least one of the players in the group

-These are unlikely to come up in multi-group grouping games (i.e. the ones with 3 or more groups)

NOT BOTH

-The necessary condition is negated

-So A-->NOT B is the same as saying you can't have both A and B (which is the same as saying NOT A or NOT B, or not both since or is inclusive)

-Shortcut: If the necessary condition is negated, you can't have both of the players in the same group

-These are the most common rule in grouping games

MUST BE TOGETHER

-Usually stated as must be together, or if and only if

-You either have both, or you have neither

-The strongest rule, and the one most likely to lead to scenarios

Tag Along

-Either neither term is negated, or both are (if both are negated, just take the contrapositive and you'll get the traditional-looking tag along)

-Will look like A-->B (or NOT B-->NOT A)

-Weakest rule, doesn't tell you much

There is one time when you can get a stronger rule than Must Be Together, and that's when you have AT LEAST ONE, BUT NOT BOTH. This can either be a single rule, or broken up into two rules. If you get this type of rule, you're golden. This means you should set up two scenarios, based on which one of your two players is in and which one is out. It makes a rough game a lot easier. Just look at the summer school grouping game, where the student has to take 4 of 8 classes (statistics at 3/9, world history, russian, japanese, geography, psychology, mathematics, and yes I did that from memory).

Thank you bp! I am so happy that I have finally crystalized it in my head - the cp you gave makes sense too I remember that summer school problem. I love the at least one but not both questions - I rip through those! Your help is much appreciated