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 Dave Killoran
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#84911
Setup and Rule Diagram Explanation

This is a Grouping/Linear Combination game.

This game features two elements: the group of four runners that will compete (Grouping), and the order of the races in which they run (Linear).

The first two rules address Linear aspects of the game, with the first rule creating a block if Q runs in the track meet (and a Q Not Law on the fourth race):

PT61_O2010 LG Explanations_g3-d1rule.png
Note that because there are only five available runners for four races, T must always run in one of the races. Otherwise, via the contrapositive, Q could not run in a race, leaving only three runners for four races.

The second rule creates two Not Laws for S (shown with the Not Law from the first rule):
PT61_O2010 LG Explanations_g3-d2rule.png
The third and fourth rules address both Grouping and Linear aspects of the game. The two rules combine to create a double-arrow rule:

PT61_O2010 LG Explanations_g3-d3rule.png
Thus, if U is not in the track meet, then R runs second, and if R runs in the second race, U cannot be in the track meet. This is a very powerful rule, because if R runs in the second race, then U cannot run in the track meet, meaning that S, Q, and T must all run in the track meet. And, because Q and T form a block, they must run in the third and fourth races, leaving S to run in the first race:

PT61_O2010 LG Explanations_g3-d4rule.png
The contrapositives of the third and fourth rules also create a double-arrow:

PT61_O2010 LG Explanations_g3-d5rule.png
Thus, if U runs in the track meet, then R cannot run second, and if R runs in any other race aside from the second race, U must be in the track meet.
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 Basia W
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#16347
Good evening,

I had a question regarding the particularities of this set-up.

In particular, I had a difficult time with the 3rd and 4th rule considering they are opposites of one another.

Third rule: Not U :dblline: Not R2 (one or the other or both appear)

Fourth: U :dblline: R2 (one or the other or both are out)

what sort of inferences could I draw with these sort of conflicting rules?

Thank you,

best,

Basia
 David Boyle
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#16348
Basia W wrote:Good evening,

I had a question regarding the particularities of this set-up.

In particular, I had a difficult time with the 3rd and 4th rule considering they are opposites of one another.

Third rule: Not U :dblline: Not R2 (one or the other or both appear)

Fourth: U :dblline: R2 (one or the other or both are out)

what sort of inferences could I draw with these sort of conflicting rules?

Thank you,

best,

Basia

Hello,

Question 3 from what?

David
 Basia W
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#16373
Oh hi sorry- it was section 3, and the set up was for 12-17
 David Boyle
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#16421
Basia W wrote:Oh hi sorry- it was section 3, and the set up was for 12-17
Basia W wrote:
Good evening,

I had a question regarding the particularities of this set-up.

In particular, I had a difficult time with the 3rd and 4th rule considering they are opposites of one another.

Third rule: Not U :dblline: Not R2 (one or the other or both appear)

Fourth: U :dblline: R2 (one or the other or both are out)

what sort of inferences could I draw with these sort of conflicting rules?

Thank you,

best,

Basia


Hello Basia,

It looks as if you actually have a biconditional here, since slash u :arrow: r2, and r2 :arrow: slash u. So,

slash u :dbl: r2.

Hope this helps,
David
 Basia W
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#16438
Hello,

Why do we cancel out the negatives in the biconditional?

thanks,

best,

Basia
 Robert Carroll
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#16557
Basia,

The negatives are not canceled out. "slash U" means "not U" in David's explanation. U is negated and R2 is not negated, since that's the way the rules are written.

Robert Carroll
 Jkjones3789
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#16601
Hello, I have a question about the third and fourth rules in this game. Are they both together supposed to be taken as a biconditional in the repeat and reverse form. If so how would one even diagram that and what would the implications of that diagram allow one to infer? Thank you !
 Nicholas Bruno
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#16608
So let's walk through this step-by-step:

First, I'd diagram rule 3: If not U :arrow: then R (sub 2).

Second, I'd diagram rule 4: If R (sub 2), then not U.

So you can see that relationship: U cannot be in the meet if R2. So I'd diagram:

Not U :dblline: R2

Does that make sense/answer your question? Please follow up if not!
 eober
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#16632
Hi,

I was just wondering if someone would be able to explain rule 2? "Smith does not run in either the 2nd or the 4th race"

Does this mean Smith cannot be placed in 2nd or 4th place but can be placed in 1st or 3rd, and he can also be absent?

Thanks!

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