Q11

[soyeonjeon]

I think A and E could be true. Can someone go through this problem?
Thanks!

Also, would "if, but only if" be G(s) --> H(r)?
That is how I solved the problem.

[hyewonkim89]

Hi Soyeon,

Below is how I solved this problem.

First of all, for your question, if and only if would make arrows that go both ways like this <--->. So 'Grace helps move the sofa if, but only if, Heather helps move the recliner' would be "G(s) <---> H(r)"

For 11, if Josh and Maria help each other move the sofa, we can infer a few things here.

First, we can infer that Heather can't move the recliner since Grace can't move the sofa. So this eliminates (A) and (B) immediately.

We can also eliminate (C) immediately since Grace and Josh can't move furniture together.

We can also infer that then Heather will help move the table since EVERY person must move at least one piece of furniture. We can eliminate (D) here since Heather has to be one of the movers for the table.

So the answer would be (E).

I hope this helps!

[christine.defenbaugh]

Fantastic explanation hyewonkim89!

The key to this question lies in understanding that the contrapositive of the biconditional is another biconditional.

G(s) <--> H(r)
NO G(s) <--> NO H(r)

For conditional questions, as always, we want to draw a new diagram to play out the conditional given in the question. Once J and M are placed in the Sofa column, we can see that we have triggered the contrapositive of the biconditional. Since G cannot be in (s) - there's no room! - that means H cannot be in (r). Since H cannot be in (s) either - there's no room! - and since H has to move something, that means H has to be in (t). Check out the diagram!



Eliminate the Violators!
hyewonkim89 is absolutely correct about the eliminations. The inference that H can never be in (r) allows us to eliminate (A) and (B). The rule that G and J can never be together eliminates (C). And finally, the inference that H must go in (t) allows us to eliminate (D) (since there wouldn't be room for both G and M in (t) alongside H)!

Who's Left?
(E) works just fine, because we can place M in (t) with H. This will force G to be in (r). M will have to join G in (r) - we already know that H can't go there, and once G is in (r), J is not allowed. See the diagram below!


Please let me know if that completely answers your question!

[samiraa180]

Christine,

I learned a lot from the previous question, and it's only fitting that I'm doing this problem, since we covered this game type in class.

I tried to use the diagram, but I wasn't able to make any deductions because of the two conditional rules. Did I miss something?

When I translate the first rule, it looks like:

G sofa---> H recliner

Contrapositive:

H sofa/ table----> G recliner/ table

I'm trying to figure out whether the contrapositive will be translated as an "or" / "and"

So, H sofa and table----> G recliner and table

OR

H sofa or table----> G recliner or table

As usual, thank you in advance.

[christine.defenbaugh]

Christine,
When I translate the first rule, it looks like: G sofa---> H recliner

Contrapositive: H sofa/ table----> G recliner/ table

I'm trying to figure out whether the contrapositive will be translated as an "or" / "and"

So, H sofa and table----> G recliner and table
OR
H sofa or table----> G recliner or table

It would be "or" in this case. However, I would caution you not to try to force negatives into positives in all cases. What I mean by that is that the biconditional rule is:

G sofa <--> H recliner

The contrapositive should look like this:

~G sofa <--> ~H recliner

If G is NOT on the sofa, then he is either on the recliner, on the table, or both. But there's a lot of ambiguity there, so I recommend just retaining the idea of "not the sofa!" by writing something like "~G sofa".

In a binary game, it would be different. If there were only the red team and the blue team, and we had a rule like this:

A red <--> B blue

the contrapositive would be: ~A red <--> ~B blue

But here, if A is NOT on the red team, he MUST be on the blue team! Using that idea, I could just re-write the contrapositive to be a little more useful:

A blue <--> B red

In this game, though, because there are three teams AND because elements can show up multiple times, it's just not as useful to try to turn the negatives into positives. "~G sofa" is more useful than writing out all the places G COULD go.

Does that help a bit?