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Since we have been working on some logical reasoning [#permalink]

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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.

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.

Necessary Conditions vs Sufficient Conditions [#permalink]

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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.

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.

Re: Necessary Conditions vs Sufficient Conditions [#permalink]

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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?

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.

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

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)"

1. My Dog Dylan loves being brushed. At the moment, he is not a happy doggie, so I can't have just brushed him.

If X, then Y => Not Y, then Not X
if I brush my Dog, he will love it => He is not happy, I can't have brushed him
CORRECT LOGIC.

2. Dylan barks loudly when he is alarmed or frightened. One night I woke up when he barked fiercely. I concluded that he must have been either alarmed or frightened, so I tiptoed down the stairs expecting to find a burglar in the house. Was my conclusion a logical one?

When Dylan is alarmed or frightened ==> Dylan barks loudly
Dylan barked loudly ==> alarmed or frightened.

Wrong Logic

Correct logic is :
If X, Then Y ==>
a) X then Y or
b) Not Y, Not X

1. Whenever the red light is on and the green light is off, it means that the protection shields are no longer in place covering the uranium core. The protection shields are covering the uranium core, yet the green light is off. This means that the red light must be off also.

Logic:
(A)
If a, then b => Not b, Not a

(B)
If Not(X and Y) ==> (X is true, Not Y) or (Y is true, Not X): Correct Logic
If Not(X and Y) ==> (X is Not true, Y true) or (Y is Not true, X): Wrong Logic

So, in the question:
(red is on) and (green is off) ==> (No Protection shield)
using (A)
(There is Protection shield) ==> Not((red is on) and (green is off))

Now we have for second Part (using B):
Not((red is on) and (green is off)) ==> (green is off), (red is off) [x, not y]
So logically Correct.

2. My biology text book tells me that no birds are mammals. I conclude that no mammals are birds.
No birds are mammals.
=> X (no birds) -> Y(mammals)
=> No Y [no (mammals)] -> No X [no (no birds)]
=> no mammals -> birds
Logically Correct.

3. Our leisure centre had a budget of Â£100,000 last year to be spent on a swimming pool costing Â£60,000 or a gymnasium costing Â£55,000. We went ahead and ordered the swimming pool to be built. Therefore we did not spend any money on having a gymnasium built last year.
Logic:
(X or Y) == Not(X and Y) => (X is true, Not Y) or (Y is true, Not X): Correct Logic
(spent 60k on swimming pool) or (spent 55k in gymnasium) => (spent 60k on swimming pool), Not (spent 55k in gymnasium)
So statement is Logically correct

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.

I didn't Understand this.
First, You said that 'if it rains, then I will take an umbrella with me.' The 'If/then' here is a condition that implies that you will take an umbrella "only" if it is raining. Does GMAT have such devious logic?

If x then y is NOT equivalent to Y only if X. In if x then y, x is the sufficient condition of y. In Y only if X, X is the necessary condition of Y.
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