#23 - Local

[Jon Denning]

Complete Question Explanation

The correct answer choice is (C).

Answer choice (A):

Answer choice (B):

Answer choice (C): This is the correct answer choice.

Answer choice (D):

Answer choice (E):


This explanation is still in progress. Please post any questions below!

C4E2EE1F-3BFF-4ED0-9159-010A005FF113 15.png

[Tami Taylor]

Hello,

Can you help me understand why the correct answer is (C) and not (D)?

I came up with two possible scenarios. Please note the base of my diagram are the names of the six corporations (running horizontally) and then the types of bond would be on top of each respective corporation, running horizontally as well:

Scenario 1: R doesn't have 10s, so L doesn't have 10s (contrapositive of 3rd rule). V and S must have both types of bonds. If V has 5s and 10s, so could S. If S has 5s and 10s, so could V (2nd rule isn't triggered by either conditional statement).
5_10_5_5_10+5_5+10
G_H_ L_R_ S _V

Scenario 2: R has 10s, so L has 10s . V has to have 5s, so that S has both types of bonds. The contrapositive of the 2nd rule says that if S doesn't have both types of bonds, then V doesn't have 5s, which means only R is the only company that can have two types of bonds. This means that the other company with two types of bonds (besides S) is either R or V.
10_5_10_10(+5)_10+5_5(+10)
G_ H_ L_ R _ S _ V

Based on this, the only condition that MUST be true is that V offers 5s.

Thanks!

[KelseyWoods]

Hi Tami!

You need to check your setup here because you've got two possibilities here, neither of which is actually possible. Pay close attention to the scenario: "Exactly four of the corporations offer 5-year bonds and exactly four offer 10-year bonds. Two of them offer both." There are six total corporations. So if four have 5-year bonds, four have 10-year bonds, and two have both, that means that we have to have exactly two corporations with 5-year bonds only, exactly two corporations with 10-year bonds only, and exactly two corporations with both 5- and 10-year bonds. That's why the base in our set up is: 5 5 10 10 B B.

Look at the diagram in the post above to see the actual possibilities for this local question. If G offers only one type of bond and H offers only the other type of bond, then we need to do two diagrams: one with H in 5 and G in 10, and the other with G in 5 and H in 10.

1) H in 5, G in 10: From the first rule, L is in 10. From the third rule, this means that R must have 10-year bonds, but since G and L are already the two corporations offering only 10-year bonds, that means that R must offer both. So then we have one corporation left offering just 5-year bonds and one offering both. From the second rule, if V offered both, S would also have to offer both. So V has to offer 5-year bonds only and S offers both.

2) G in 5, H in 10: From the first rule, L is in 5. So of R V S, one of them has only 10-year bonds and the other two have both. Based on the second rule, S has to have both: if V had only 10-year bonds, both S and R would have both; if V had both bonds, S would have to have both as well. So either way, S has both. For R and V, one of them will have only 10-year bonds and the other will have both.

Based on those possibilities, R must always offer 10-year bonds--it offers both in possibility 1 and 10 only or both in possibility 2. V does not always offer 5-year bonds, though, because in possibility 2 it can offer 10-year bonds only.

Hope this helps!

Best,
Kelsey

[Tami Taylor]

Hi Kelsey,

Yes, this really helps. Thank you for going into so much detail!

[itsmathuri]

Hey there!

I think I'm either incorrectly interpreting a rule or missing a major inference. I'm not sure how to account for R, S, and V. Could you please explain how you got to the correct answer?

Thanks in advance!

[Robert Carroll]

mathuri,

Given our base (viewtopic.php?p=47466#p47466), the local condition in this question creates two situations:

G is 5 only, H is 10 only

G is 10 only, H is 5 only

In the first situation, L must be 5 only, via the first rule. Note that means that G and L are 5 only - nothing else is 5 only, so everything else is 10 or both. Well, R is still unassigned, so R is 10 only or both, meaning answer choice (C) is true in this situation.

In the second situation, L is 10, via the first rule, and now the third rule triggers and forces R to be 10 or both. That's enough to show that, again, answer choice (C) is true.

So no matter how G and H are split up, answer choice (C) is true, so it must be true.

Robert Carroll

[bluebanana36]

Hi,
The third rule is L10 --> R10; and would the contrapositive be not R10 (so R5) --> not L10 (so L5).
If that's the case wouldn't in the scenario where we have:
5: HV
10: GL
Both: RS

Wouldn't this not work because if R5, then L5; and R sells 5 year bonds here (in addition to the 10), but L does not sell 10 year bonds?

Thank you very much!

[Robert Carroll]

blue,

Jeremy's post here: viewtopic.php?p=76802#p76802 explains the mistake in your contrapositive.

In order to replace one statement by another, they must be logically equivalent. So start with the last rule:

L10 R10

We can, as always, contrapose that:

R10 L10

Well, if R10, then R5, so we're tempted to replace the sufficient condition of the contrapositive by "R5". But that's not accurate.

R10 R5

is true, but

R5 R10

is not.

So R10 and R5 are not logically equivalent to each other.

It makes sense why this doesn't work. In your hypothetical, R sells both types of bonds. So R10 is false in that hypothetical; R is selling both types of bonds, and thus selling 10-year bonds. So the contrapositive is not relevant; the sufficient condition of the contrapositive is R10, but that's not true, so the contrapositive does nothing.

This failure of the equivalence of "Not value 1" to "Is value 2" in an apparently two-valued game came come about, as here, because something can possess both values, or, in a different game, if something might possess neither value. Because every company offers at least one type of bond, we don't get the latter problem here, but we do get the former problem.

Robert Carroll

[kupwarriors9]

If V offers both bonds, does that mean that S will be required to offer both BECAUSE V is offering 5-year bonds and it's the necessary condition? Or, if V offers both bonds can S only solely offer 5yb or 10yb?

mathuri,

Given our base (viewtopic.php?p=47466#p47466), the local condition in this question creates two situations:

G is 5 only, H is 10 only

G is 10 only, H is 5 only

In the first situation, L must be 5 only, via the first rule. Note that means that G and L are 5 only - nothing else is 5 only, so everything else is 10 or both. Well, R is still unassigned, so R is 10 only or both, meaning answer choice (C) is true in this situation.

In the second situation, L is 10, via the first rule, and now the third rule triggers and forces R to be 10 or both. That's enough to show that, again, answer choice (C) is true.

So no matter how G and H are split up, answer choice (C) is true, so it must be true.

Robert Carroll

[Robert Carroll]

kup,

If V offers both types, will V offer 5-year bonds? Yes; so the conditional triggers, and S must offer both as well.

Robert Carroll