Oooh - good point about G... I failed to account for falling together.
For H, I think it's important to use more than just OR - as you said, both D and Y can be true, and the statement (D ⋁ Y) will be true. I guess the problem doesn't explicitly state that the agreement will be good for ONLY one side, but it does seem to imply that it won't be good for both.
the latter one doesn't fit since they both must be negated ... assuming the question is implying either/or, that only one will remain strong. If both are assumed to become weak instead, then it does fit.
THEN the opposite results or fit occur for the OR statement, according to which assumption you use.
I'm having a hard time with some of this. For D I put T=> (D^⌐ Y)V(Y^⌐ D) which gives either D or Y strong, but what about the possibility with V that both can be strong? I'm having the same problem with V in G and H. Back to D, it also seems possible that you can have T ^ ⌐D ^⌐ Y.
(d) If new trade agreement is signed, Dollar and Yuan can't both remain strong.
I think "can't both be strong" allows for both to be weak. T=> (D ^ ~Y) v (Y ^ ~D) does not allow both to be weak. Both T => ~D v ~Y and T => ~(D ^ Y) require only one of them to be weak and allow both to be weak.
I worked through the tutorial at http://www.millersville.edu/~bikenaga/math-proof/truth-tables/truth-tables.html and that helped a lot. For G I got: T=>~(Y^D) Does this work? For H I got T=>(DVY) ^ ~(D^Y)
For G, if they both rise together, that would be T => (D^Y), and if they both fall together, that would be T=> (~D^~Y). T => ~(Y^D) would say only one may rise or both may fail.
Your H is correct if you infer "New trade agreement will be good for [only] one side, but no one knows which."
Here are my answers:
ReplyDelete(a) New trade agreement will lead to strong currencies in both countries.
T ⇒ D ⋀ Y
(b) Strong Dollar means a weak Yuan
D ⇒ ¬Y
(c) Trade agreement fails on news of weak Dollar.
¬D ⇒ ¬T
(d) If new trade agreement is signed, Dollar and Yuan can't both remain strong
T ⇒ ¬(D ⋀ Y)
(e) Dollar weak but Yuan strong, following new trade agreement.
T ⇒ ¬D ⋀ Y
(f) If the trade agreement is signed, a rise in the Yuan will result in a fall in the Dollar.
T ⋀ Y ⇒ ¬D
(g) New trade agreement means Dollar and Yuan will rise and fall together.
T ⇒ D ⋀ Y
(h) New trade agreement will be good for one side, but no one knows which.
T ⇒ (D ⋁ Y) ⋀ ¬(D ⋀ Y)
For (g) I have T => (D ^ Y) v (~D ^ ~Y)
DeleteOn problem D, we have two answers here:
ReplyDeleteT => ~D V ~Y
T ⇒ ¬(D ⋀ Y)
Are these equivalent? Let's have a discussion...
My truth table says these are the same:
DeleteD | ⌐D |Y | ⌐Y | D ∩Y |⌐(D ∩Y)| ⌐ D V⌐ Y
T |F |T |F |T |F |F
T |F |F |T |F |T |T
F |T |T |F |F |T |T
F |T |F |T |F |T |T
That's exactly what I did to confirm this. :)
DeleteMax I think they are equivalent, just like you distribute the minus sign among the contents in parentheses.
ReplyDeleteMy G answer is T => (D and Y) and -(D and Y)
My H answer is T => D V Y since OR can have true or false.
Very interesting in comparison to yours.
(I don't know how to get the same math symbols on my keyboard)
Nancy Colbert
wait a minute: just noticed Max' two answers are very different because of and, or. gotta go think about it
ReplyDeleteOooh - good point about G... I failed to account for falling together.
ReplyDeleteFor H, I think it's important to use more than just OR - as you said, both D and Y can be true, and the statement (D ⋁ Y) will be true. I guess the problem doesn't explicitly state that the agreement will be good for ONLY one side, but it does seem to imply that it won't be good for both.
no they are not equivalent I decided:
ReplyDeletethe latter one doesn't fit since they both must be negated ... assuming the question is implying either/or, that only one will remain strong. If both are assumed to become weak instead, then it does fit.
THEN the opposite results or fit occur for the OR statement, according to which assumption you use.
you are right about H --- how quickly I lose all the details of the truth table!!
ReplyDeleteexcept I'm not convinced yet about your AND in the middle...question just popped up....to be continued
ReplyDeleteI'mm on a roll:
ReplyDeleteH: T => (D and - Y) AND ( -D and Y)
all "ands", it turns out :)
Still, I'm wondering if I missed something again!
I'm having a hard time with some of this. For D I put
ReplyDeleteT=> (D^⌐ Y)V(Y^⌐ D)
which gives either D or Y strong, but what about the possibility with V that both can be strong? I'm having the same problem with V in G and H.
Back to D, it also seems possible that you can have T ^ ⌐D ^⌐ Y.
(d) If new trade agreement is signed, Dollar and Yuan can't both remain strong.
DeleteI think "can't both be strong" allows for both to be weak. T=> (D ^ ~Y) v (Y ^ ~D) does not allow both to be weak. Both T => ~D v ~Y and T => ~(D ^ Y) require only one of them to be weak and allow both to be weak.
I worked through the tutorial at http://www.millersville.edu/~bikenaga/math-proof/truth-tables/truth-tables.html and that helped a lot.
DeleteFor G I got: T=>~(Y^D) Does this work?
For H I got T=>(DVY) ^ ~(D^Y)
For G, if they both rise together, that would be T => (D^Y), and if they both fall together, that would be T=> (~D^~Y). T => ~(Y^D) would say only one may rise or both may fail.
DeleteYour H is correct if you infer "New trade agreement will be good for [only] one side, but no one knows which."