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Since we have been working on some logical reasoning

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Since we have been working on some logical reasoning [#permalink] New post 02 Mar 2005, 09:35
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Since we have been working on some logical reasoning questions I'm going to try to collect the principles I'm following here for everybody's reference. Please feel free to discuss and add more. :)

If X then Y
This is the equivalent of: If non Y then non X.
Example: If it rains, then I will take an umbrella with me. I don't have a umbrella with me. That must mean it is not raining.

This is NOT equivalent to: If Y then X, or If Y then non X, or if non Y then X. In fact, if we know "If X then Y" and Y occurred, X may or may not happen.
Example. If it rains, then I will definitely take an umbrella with me. I have a umbrella with me today. Is it raining? It may or may not be raining. I said if it rains I will take an umbralla with me. But I could also take an umbralla with me just for the sake of it, even if it doesn't rain. By the same token, if it is not raining, do I have an umbralla with me? I may or may not have.

Using symbals:
X->Y<nonY>nonX
These two below are the same thing:
nonX->Y<nonY>X
X->nonY <Y>non X

Y if and only if X
This is the equivalent of: If X then Y, AND if Y then X. Also, if non X then non Y. If non Y then non X.

Example:
I will take an umbralla with me if and only if it rains. If it rains, then I have the umbralla with me. If I have the umbralla, then it must be raining. If I don't have the umbralla, then it mustn't be raining. If it isn't raining, then I don't have the umbralla with me.

Y unless X
This is the equivalent of: If non X then Y. Also, if non Y then X.

Example:
I will take an umbralla with me unless it is sunny. If it is not sunny, I will take an umbralla with me. If I don't have an umbralla with me, it must mean that it is sunny. However, if it is sunny, I may or may not take an umbralla with me. If I have my umbralla with me, it may or may not be sunny.

Last edited by HongHu on 30 Mar 2005, 08:56, edited 4 times in total.
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Necessary Conditions vs Sufficient Conditions [#permalink] New post 03 Mar 2005, 10:19
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Necessary conditions:

If A is a necessary condition of B, that means A must happen for B to happen. In other words, if B happened, A must be true. If A is not true, then B can't happen.

In summary: If B then A. If non A then non B.

Example: I will take my umbralla with me only if it rains.
Raining is a necessary condition for my taking the umbralla with me.
If it is not raining, you can be sure that I don't have my umbralla with me.

Sufficient conditions:

If A is a sufficient condition of B, that means if A happens B must happen. In other words, if B did not happen, A must be false.

In summary: If A then B. If non B then non A.

Example: I will take my umbralla with me if it rains.
Raining is a sufficient condition for me to take the umbralla with me.
If it is not raining, you are not sure whether I have my umbralla with me. But if I don't have my umbralla with me, you can be sure that it is not raining.

Necessary and sufficient
If A is a necessary and sufficient condition for B, that means if A happen B must happen, and if A does not happen, B does not happen. In other words, A=B. It is equivalent with B if and only if A.

In summary: If A then B. If B then A. If non B then non A. If non A then non B.

Example: I will take my umbralla if and only if it rains.
If it is raining, you can be sure I have the umbralla. If it is not raining, I don't have the umbralla. If I have my umbralla with me, you can be sure that it is raining. If I don't have it with me, it mustn't be raining.
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 [#permalink] New post 20 Jun 2006, 05:25
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Futuristic provided me with another good example for logic type CRs.

Futuristic wrote:
Not all tenured faculty are full professors. Therefore, although every faculty member in the linguistics department has tenure, it must be the case that not all of the faculty members in the linguistics department are full professors.

The flawed pattern of reasoning exhibited by the argument above is most similar to that exhibited by which one of the following?
(A) Although all modern office towers are climate-controlled buildings, not all office buildings are climate-controlled. Therefore, it must be the case that not all office buildings are modern office towers.
(B) All municipal hospital buildings are massive, but not all municipal hospital buildings are forbidding in appearance. Therefore, massive buildings need not present a forbidding appearance.
(C) Although some buildings designed by famous architects are not well proportioned, all government buildings are designed by famous architects. Therefore, some government buildings are not well proportioned.
(D) Not all public buildings are well designed, but some poorly designed public buildings were originally intended for private use. Therefore, the poorly designed public buildings were all originally designed for private use.
(E) Although some cathedrals are not built of stone, every cathedral is impressive. Therefore, buildings can be impressive even though they are not built of stone.


The first step I did is to symbolize the stem:
Not all T are F. All L are T. Therefore not all L are F.

The correct conclusion is that it may be true that not all L are F, but it is equally true that all L are F. If we know that all T are L, but not all T are F, then we know for sure not all L are F.

It may be helpful to think the other way. Some Ts are not F. All L are T. But all Ls may be the Ts that are F, or they may be the Ts are not F, we don't know.

So now we know what the error is, then we can proceed.

Quote:
(A) Although all modern office towers are climate-controlled buildings, not all office buildings are climate-controlled. Therefore, it must be the case that not all office buildings are modern office towers.

All M are C. Not all O are C. Therefore not all O are M.
Compare this with the stem.
All L are T. Not all T are F. Therefore not all L are F.
Note L corresponds to M and T corresponds to C. The second condition is not the same.

Quote:
(B) All municipal hospital buildings are massive, but not all municipal hospital buildings are forbidding in appearance. Therefore, massive buildings need not present a forbidding appearance.

All H are M. Not all H are F. Therefore not all H are F.
Again you see the flow is different. Both first and second condition starts with H, while in stem it flows like this: L->T->F.

Quote:
(C) Although some buildings designed by famous architects are not well proportioned, all government buildings are designed by famous architects. Therefore, some government buildings are not well proportioned.

Not all F are W, all G are F. Therefore not all G are W.
Stem:
Not all T are F. All L are T. Therefore not all L are F.
Exactly the same flow.

Quote:
(D) Not all public buildings are well designed, but some poorly designed public buildings were originally intended for private use. Therefore, the poorly designed public buildings were all originally designed for private use.

Not all P are W. Some nonW are U. Therefore all nonW are U.
Not same, don't you think?

Quote:
(E) Although some cathedrals are not built of stone, every cathedral is impressive. Therefore, buildings can be impressive even though they are not built of stone.

Some C are nonS. All C are I. Therefore some nonS can be I.
Again not the same logic.

For this type of questions it is very helpful to symbolize it, then you can ignore the contents and simply compare the logic structure. For this question you don't even need to understand the logic and find the error. Simply translation and comparation would be sufficient. However, it is often very helpful if you could understand why a logic is fause for logical questions.
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 [#permalink] New post 20 Jun 2006, 05:34
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Another good example where logic can be applied:

Quote:
Brown dwarfs—dim red stars that are too cool to burn hydrogen—are very similar in appearance to red dwarf stars, which are just hot enough to burn hydrogen. Stars, when first formed, contain substantial amounts of the element lithium. All stars but the coolest of the brown dwarfs are hot enough to destroy lithium completely by converting it to helium. Accordingly, any star found that contains no lithium is not one of these coolest brown dwarfs.
The argument depends on assuming which one of the following?
(A) None of the coolest brown dwarfs has ever been hot enough to destroy lithium.
(B) Most stars that are too cool to burn hydrogen are too cool to destroy lithium completely.
(C) Brown dwarfs that are not hot enough to destroy lithium are hot enough to destroy helium.
(D) Most stars, when first formed, contain roughly the same percentage of lithium.
(E) No stars are more similar in appearance to red dwarfs than are brown dwarfs.


Fact: BD are similar to RD in appearance but RD can burn hydrogen and BD cannot.
Fact:Stars have lithium when first formed.
Fact: All stars but the coolest BD can destroy lithium completely.
Conclusion: If there is no lithium, it is not a coolest BD.

Ask for assumption.
Looking at the facts the first one doesn't do much except to tell us BD is cool. Two and three are important. The logic here is that all NonC(oolest BD) can do D(estroy lithium). Therefore if D it is nonC. Obviously the logic is wrong in saying if A is B then B is A. We know that all nonC can D, and some C cannot D, but perhaps some C can D also? Now look at the options.

Quote:
(A) None of the coolest brown dwarfs has ever been hot enough to destroy lithium.

Aha, exactly what we are looking for. Obviously it is assuming that none of the C can Destroy lithium, so that we can reach the conclusion. (If all A is B, and none B is nonA, then B is A.)

You then look at the rest of the options, and make sure there isn't something that is equally plausible. If so then you need to reread everything again.

Quote:

(B) Most stars that are too cool to burn hydrogen are too cool to destroy lithium completely.

We already know that hydrogen is irrelevent here.
Quote:
(C) Brown dwarfs that are not hot enough to destroy lithium are hot enough to destroy helium.

Obviously helium is irrelevent also.
Quote:
(D) Most stars, when first formed, contain roughly the same percentage of lithium.

This one is a little curious, but since we already formed our answer, we know this is not it. It could be the answer if the question stem is different. Say if the conclusion is that "It must be a coolest BD if there are lithium in the star." Then the right answer may very well be this one, since you would be assuming all D process should have been completed when you found the star, otherwise there may be some lithrium left undestroyed when you found a star even if it's not the coolest BD.
Quote:

(E) No stars are more similar in appearance to red dwarfs than are brown dwarfs.

Again we know this is irrelevant. It can be rather quick to go through the options if you already know what you are looking for.
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Re: If X then Y [#permalink] New post 02 Mar 2005, 12:18
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Good job....to put this in our heads mathematically, u can deduce these.


----> just like we inverse inequality sign when we multiply -ve, we can perhaps do the same for X---> Y

1) For X--> Y......-ve on both sides will give.....(non Y ----> non X)...note the relation is reversed i.e. Y to X now

2) For (non X ----> Y)......multiply-ve on both sides.....(non Y ----> X)

3) For (X----> non Y).....multiply-ve on both sides.....(Y-----> non X)

Just a thought on mathematical approach for easier recollection.

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More on Some and All [#permalink] New post 18 Aug 2005, 06:50
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Paul has made a very good example to show the relationship between some and all.
Paul wrote:
Which of the following conclusions can be deduced from the two statements below?

Some Alphas are not Gammas
All Betas are Gammas

A) Some Alphas are not Betas
B) No Gammas are Alphas
C) All Gammas are Betas
D) All Alphas are Gammas
E) Some Alphas are Gammas


What we know:
Some A are not G. And all B are G.

What can we infer from here?
Very limited. If it is an A then it may or may not be G. If it is B then it is definitely G. If it is a G then it may or may not be A, and it may or may not be B. If it is not G then it must not be a B.

What can we NOT infer from here?

Some A are G. -- That's very plausible, but very WRONG! We know that some A are not G, but it is possible that all A are not G, we just don't know. For example: some snakes do not have feet. From this statement, can we conclude that some snakes do have feet? NO.

By the same token, we can't say some G are A. We don't know if some animals with feet are snakes.

Some G are not A. -- Again this is wrong. Although some A are not G, but perhaps G only include the As that is G. Say some integers are odd numbers. Obviously it would be wrong to say some odd numbers are not integers.

All G are B. -- This is pretty obvious. All odd numbers are integers but not all integers are odd.

Now lets look at the statement one by one.

A) Some Alphas are not Betas
We know that some A are not G. And if it is not G then it must not be B. Therefore some A are not B.
Correct

B) No Gammas are Alphas
We konw some A are not G. It is possible that some A are G, and it is also possible that no A is G. We just don't know. If no A is G then no G is A, but if some A are G then some G must be A. Some integers are not odd, but it is incorrect to say that no odd numbers are integers.
In other words this is NOT always true.

C) All Gammas are Betas
We know that all B are G, but we don't know if all G are B. All odd numbers are integers but not all integers are odd.

D) All Alphas are Gammas
Obviously wrong because we know some A are not G.

E) Some Alphas are Gammas
We don't know about it. It may or may not be true. Perhaps some A are G. Perhaps no A is G. All we know is some A are not G.
Say A=integers, G=odd numbers. Some integers are not odd numbers, but some integers are odd numbers.
Now let's say A=even numbers, G=odd numbers. Some even numbers are not odd numbers. But it will be wrong to say some even numbers are odd numbers.
The key here is when we make a statement about "some", we are not saying anything about the rest. The rest may or may not be different from the "some" that we have made a statement about.
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More discussions on if ... then ... [#permalink] New post 19 Aug 2005, 08:29
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More discussions on if ... then ...

Question:
If i say X happens only because of Y is that the same as saying X if and only if Y

Answer:
No. X happens only because Y happened, in other words, if Y does not happen then X will not happen. That means X->Y. If X happened, we know for sure that Y has happened. But we don't know if Y happens whether X will happen. So it is only half of the "if and only if" condition. Y is necessary for X to happen, but we don't know if it is sufficient.

Question:
I Agree that X happens only because of Y means X--->Y, that takes care of the "only if condition". Now in this example:

It gets "chilly in summer" because of the "unexpected cold front"

Does it not mean
unexpected cold front ----> chilly in summer

All I’m doing after this point is adding an "only " clause to it

Answer:
Yes the word "only" makes all the difference.

If you say X happens because of Y, then it means Y->X. It gets cold because the cold front moves in. If cold front, then it gets cold. Other things may lead to X too. When we see X it doesn't mean Y must be there. Winter makes it cold too, even without a cold front moving in. So when you feel that it is getting cold, you can't say that there must be a cold front moving in.

If you say X happens "only" because of Y, then it means X->Y. If you say that it only gets cold when there is a cold front, then whenever it gets cold, you know that there is a cold front. What you don't know from that statement though, is whether everytime when the cold front moves in it gets cold.

"if" and "only if" is the formal usage. "because" is simply common language, so you need to carefully read the sentence and translate it into the logistic languages. I do not think there is an equivalent for "if and only if" using "because", but there will be many other common languages that would express the same thing.

Question:
One more time. X is the only cause of Y. Can you throw more light on this. What does it mean and what does it not mean?

Answer:
Only X causes Y, no others causes Y. When we see Y we know X must be there. Y->X.

I don't think X causes Y every time though, so we can't say X->Y.

eg. Lacking food is the only cause of starving to death. If somebody is starved we know he must be in lack of food. But if somebody lacks food he might not be starved to death.
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 [#permalink] New post 02 Mar 2005, 11:52
This is sticky material!!.
Abstracting the info given in the stem is the way to attack the formal logic/paraellism questions. I was trying to keep all the information straight in my head and do the saurya's campaign stop q under 2 mins and got it wrong. I will internalize this, thanks.
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Re: If X then Y [#permalink] New post 02 Mar 2005, 12:26
Good strategy! I'm making this sticky.
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 [#permalink] New post 02 Mar 2005, 20:03
Great post Honghu, it is often not easy to verbalize such abstract concepts :good
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 [#permalink] New post 03 Mar 2005, 11:27
This is seriously an awesome post, our forum here is worth more than any kaplan course!:read
You guys are awesome!
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 [#permalink] New post 05 Mar 2005, 17:27
Thanks Honghu, just what I was looking for ever since we've been working on logical reasoning CR question types, recently. :cool
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Always, Never, and sometimes [#permalink] New post 07 Mar 2005, 08:43
We have to be careful about sometimes, always, and never.

If A = always doing something
then non A = not doing something sometimes

If A = never doing something
then non A = doing something sometimes

If A = doing something sometimes
then non A = never doing something

If A = not doing something sometimes
then non A = always doing something

For example:

Birds sing sometimes. A never sings. Therefore A is not a bird.
Birds don't sing sometimes. A always sings. Therefore A is not a bird.

Compare to:

Birds always sing. A doesn't sing sometimes. Therefore A is not a bird.
Birds never sing. A sings sometimes. Therefore A is not a bird.

Compare to:
Birds sing sometimes. A sings sometimes. Is A a bird? We don't know. A may be a person who sings sometimes.
Birds sing sometimes. A doesn't sing sometimes. Is A not a bird? We don't know. A maybe a bird who sings sometimes and doesn't sing the other times.
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 [#permalink] New post 07 Mar 2005, 15:47
Hong Thanks for the post its really awesome!!!
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 [#permalink] New post 13 Mar 2005, 11:38
Oh this is a gr8 post :circle
Thx Hong...
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Re: Necessary Conditions vs Sufficient Conditions [#permalink] New post 21 Mar 2005, 00:42
HongHu wrote:
Necessary conditions:

If A is a necessary condition of B, that means A must happen for B to happen. In other words, if B happened, A must be true. If A is not true, then B can't happen.

In summary: If B then A. If non A then non B.

Example: I will take my umbralla with me only if it rains.
Raining is a necessary condition for my taking the umbralla with me.
If it is not raining, you can be sure that I don't have my umbralla with me.

.


Hi, HongHu, first thanks for your great post.
I have a question. We often heard about a necessary condition is not a sufficient condition.

Does that mean it rains, but I may take or not take an umbrella?
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 [#permalink] New post 21 Mar 2005, 05:50
Yes, that's very good. If raining is the necessary condition of me bringing the umbrella, then if I have the umbrella you know that it must be raining. However, if it is raining, I may or may not take the umbrella with me.
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 [#permalink] New post 25 Mar 2005, 00:27
In the admissions context:

An undergraduate degree is a necessary condition for admission to Business School X

however

Holding an undergrad degree is not a sufficient condition for admission to School X since only 20% of applicants with undergraduate degrees are accepted each year
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Re: If X then Y, Help for CR [#permalink] New post 29 Mar 2005, 23:42
HongHu wrote:
Since we have been working on some logical reasoning questions I'm going to try to collect the principles I'm following here for everybody's reference. Please feel free to discuss and add more. :)

If X then Y
This is the equivalent of: If non Y then non X.
Example: If it rains, then I will take an umbrella with me. I don't have a umbrella with me. That must mean it is not raining.

This is NOT equivalent to: If Y then X, or If Y then non X, or if non Y then X. In fact, if we know "If X then Y" and Y, X may or may not happen.
Example. If it rains, then I will definitely take an umbrella with me. I have a umbrella with me today. Is it raining? It may or may not be raining. I said if it rains I will take an umbralla with me. But I could also take an umbralla with me just for the sake of it, even if it doesn't rain. By the same token, if it is not raining, do I have an umbralla with me? I may or may not have.

Using symbals:
X->Y<=>nonY->nonX
These two below are the same thing:
nonX->Y<=>nonY->X
X->nonY <=>Y->non X

Y if and only if X
This is the equivalent of: If X then Y, AND if Y then X. Also, if non X then non Y. If non Y then non X.

Example:
I will take an umbralla with me if and only if it rains. If it rains, then I have the umbralla with me. If I have the umbralla, then it must be raining. If I don't have the umbralla, then it mustn't be raining. If it isn't raining, then I don't have the umbralla with me.

Y unless X
This is the equivalent of: If non X then Y. Also, if non Y then X.

Example:
I will take an umbralla with me unless it is sunny. If it is not sunny, I will take an umbralla with me. If I don't have an umbralla with me, it must mean that it is sunny. However, if it is sunny, I may or may not take an umbralla with me. If I have my umbralla with me, it may or may not be sunny.


Hi Honghu,
for the if x then y section, you have this sentence: "This is NOT equivalent to: If Y then X, or If Y then non X, or if non Y then X. In fact, if we know "If X then Y" and Y, X may or may not happen. "

I think you meant: "In fact, if we know "If X then Y", then X may not or may not happen if Y occurs (y->not sure)"

Is that right ?
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 [#permalink] New post 30 Mar 2005, 08:55
Yes, I meant If X then Y and if we know that Y occurs, X may or may not happen. I'll add the word "occurs" in the text for easy understanding. Thanks.
  [#permalink] 30 Mar 2005, 08:55
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Since we have been working on some logical reasoning

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