Logical reasoning question 2

[Qwertyuiop[]]

Hi, i have another question. I made a diagram like this to more easily understand the statements in the question. So I made arrows to indicate who depends on who. If arrow from Derek points to Cheryl that means Cheryl goes if Derek goes. However, even after doing all this , I don't know what option to choose : P . The only option I am sure is wrong is a) If Arnold goes, Cheryl will also go. Arnold and Cheryl are not related with arrows so Cheryl doesn't depend on Arnold, he only goes if Derek goes. And not sure about the other options. Is my approach to this question correct ??

 

[blamocur]

Some of the arrows point in the wrong direction.

 

[Dr.Peterson]

Hi, i have another question. I made a diagram like this to more easily understand the statements in the question. So I made arrows to indicate who depends on who. If arrow from Derek points to Cheryl that means Cheryl goes if Derek goes. However, even after doing all this , I don't know what option to choose : P . The only option I am sure is wrong is a) If Arnold goes, Cheryl will also go. Arnold and Cheryl are not related with arrows so Cheryl doesn't depend on Arnold, he only goes if Derek goes. And not sure about the other options. Is my approach to this question correct ??

I would draw arrows differently, so that when it says if C goes, then D goes, I would make an arrow from C to D. That makes more sense logically, representing implication rather than mere dependence. In any case, your arrows are not consistent with what you say.

Think carefully about the second statement, in which the "dependence" works differently than the others.

 

[pka]

I would encourage you to write this as traditional logic symbols:

1. [imath]~C \Rightarrow D[/imath]
2. [imath]~\neg A \Rightarrow ~\neg D\text{ same as } D \Rightarrow A[/imath]
3. [imath]~A \Rightarrow B[/imath]

That mean [imath]~C \Rightarrow D\Rightarrow A\Rightarrow B\text{ same as } \neg B \Rightarrow \neg C[/imath]
What is the correct conclusion?

 

[Qwertyuiop[]]

I would encourage you to write this as traditional logic symbols:

1. [imath]~C \Rightarrow D[/imath]
2. [imath]~\neg A \Rightarrow ~\neg D\text{ same as } D \Rightarrow A[/imath]
3. [imath]~A \Rightarrow B[/imath]

That mean [imath]~C \Rightarrow D\Rightarrow A\Rightarrow B\text{ same as } \neg B \Rightarrow \neg C[/imath]
What is the correct conclusion?

Hmm so if B doesn't go C doesn't go . That makes option d) the answer which is correct! You made it look too easy. What was I doing wrong? Arrows pointing in the wrong direction lol? The 2nd statement is presented differently and that confused me maybe.
I wasn't able to draw any conclusion between B and C, if you look at my diagram.. What is the meaning of "¬" in the 2nd statement? First time I've seen them .

 

[Qwertyuiop[]]

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I would draw arrows differently, so that when it says if C goes, then D goes, I would make an arrow from C to D. That makes more sense logically, representing implication rather than mere dependence. In any case, your arrows are not consistent with what you say.

Think carefully about the second statement, in which the "dependence" works differently than the others.

Is the 2nd statement like a negation of a statement? I interpreted "If Arnold doesn't go , Derek won't go either" as "If Arnold goes, Derek goes" so that all 3 statements match. Are these two equivalent?

 

[pka]

Hmm so if B doesn't go C doesn't go . That makes option d) the answer which is correct! You made it look too easy. What was I doing wrong? Arrows pointing in the wrong direction lol? The 2nd statement is presented differently and that confused me maybe.
I wasn't able to draw any conclusion between B and C, if you look at my diagram.. What is the meaning of "¬" in the 2nd statement? First time I've seen them .

Is the 2nd statement like a negation of a statement? I interpreted "If Arnold doesn't go , Derek won't go either" as "If Arnold goes, Derek goes" so that all 3 statements match. Are these two equivalent?

Frankly. I have no clue as to what course you are doing?
However, this is a standard type of question in a course on logic.
Each of the three statements you posted is in the [imath]\bf\text{If }P\text{ then }Q\text{: form }P \Rightarrow Q[/imath].
The following are well known equivalences: [imath]\left( {P \Rightarrow Q} \right) \equiv \left( {\neg P \vee Q} \right) \equiv \left( {\neg Q \Rightarrow \neg P} \right)[/imath].
Moreover, [imath]\left[ {\left( {P \Rightarrow Q} \right) \wedge \left( {Q \Rightarrow R} \right)} \right] \Rightarrow \left( {P \Rightarrow R} \right)[/imath]i.e. implication is transitive.
Thus (d.) is the correct answer.

[imath][/imath][imath][/imath][imath][/imath]

 

[Qwertyuiop[]]

Frankly. I have no clue as to what course you are doing?
However, this is a standard type of question in a course on logic.
Each of the three statements you posted is in the [imath]\bf\text{If }P\text{ then }Q\text{: form }P \Rightarrow Q[/imath].
The following are well known equivalences: [imath]\left( {P \Rightarrow Q} \right) \equiv \left( {\neg P \vee Q} \right) \equiv \left( {\neg Q \Rightarrow \neg P} \right)[/imath].
Moreover, [imath]\left[ {\left( {P \Rightarrow Q} \right) \wedge \left( {Q \Rightarrow R} \right)} \right] \Rightarrow \left( {P \Rightarrow R} \right)[/imath]i.e. implication is transitive.
Thus (d.) is the correct answer.

[imath][/imath][imath][/imath][imath][/imath]

"Frankly. I have no clue as to what course you are doing?"
I am preparing for a university entrance test for a mechanical engineering degree. And I believe these are some questions that appeared in the test in previous sessions. I really had no clue how to go about solving these so I came and posted here. Are there any resources i can find online to help me prepare for the test? Down below I have attached the "Guide to the test" I received from the uni and this pic outlines the topics you can get in Logical reasoning questions. I have no idea what are those. Also is there a more detailed guide on the property/properties you quoted in your answer that i can find on the web? thanks

 

[Dr.Peterson]

Is the 2nd statement like a negation of a statement? I interpreted "If Arnold doesn't go , Derek won't go either" as "If Arnold goes, Derek goes" so that all 3 statements match. Are these two equivalent?

The proper term is not "negation", but "contrapositive". I presume you haven't learned that, because you are not studying logic in the formal sense, but are expected to think through things on your own. ("Logical reasoning" is a term that can cover ideas at many levels.)

Here's the idea: You know that if A doesn't happen, then D doesn't. If you are told that A does happen, you can't be sure of anything; you aren't told what happens in that case. But if you are told that D does happen, you know that A can't must have happened. right? So this statement is equivalent to "If D happens, then A happens". [edited]

"Frankly. I have no clue as to what course you are doing?"
I am preparing for a university entrance test for a mechanical engineering degree. And I believe these are some questions that appeared in the test in previous sessions. I really had no clue how to go about solving these so I came and posted here. Are there any resources i can find online to help me prepare for the test? Down below I have attached the "Guide to the test" I received from the uni and this pic outlines the topics you can get in Logical reasoning questions. I have no idea what are those. Also is there a more detailed guide on the property/properties you quoted in your answer that i can find on the web? thanks

Now you have told us the context (which would have been helpful from the start). You are being told that although you don't need specific knowledge of formal logic (such as the term "contrapositive", or the symbols pka used), it would be good to study the basics of logic. You might start by looking for an online course or textbook that teaches some of the terms used there. I'll try looking for one that seems like the right level, or perhaps others will.

 

[blamocur]

Here's the idea: You know that if A doesn't happen, then D doesn't. .... But if you are told that D does happen, you know that A can't have happened.

Looks like a typo to me. [fixed it]

 

[Dr.Peterson]

Here's the idea: You know that if A doesn't happen, then D doesn't. If you are told that A does happen, you can't be sure of anything; you aren't told what happens in that case. But if you are told that D does happen, you know that A must have happened. right? So this statement is equivalent to "If D happens, then A happens".

Yes, I was typing in a hurry.

 

[Cubist]

Are you familiar with truth tables? If so, then the following might help (1=true, 0=false)

Truth table for implies

X Y | X=>Y
====|=====
0 0 |  1
0 1 |  1
1 0 |  0  ...this line is false, since X is NOT implying Y.
1 1 |  1     All the other lines are not in conflict with the "=>"

Now you can use the above truth table to help show that not A => not D is the same as D => A
Let X₁=not A, Y₁=not D
Let X₂=D, Y₂=A

     |         not A=>not D  |           D=>A
A D  | X1 Y1       X1=>Y1     | X2 Y2    X2=>Y2
=====|=======================|=================
0 0  |  1 1         1        |  0 0       1    
0 1  |  1 0         0        |  1 0       0    
1 0  |  0 1         1        |  0 1       1    
1 1  |  0 0         1        |  1 1       1

 

[Qwertyuiop[]]

Are you familiar with truth tables? If so, then the following might help (1=true, 0=false)

Truth table for implies

X Y | X=>Y
====|=====
0 0 |  1
0 1 |  1
1 0 |  0  ...this line is false, since X is NOT implying Y.
1 1 |  1     All the other lines are not in conflict with the "=>"

Now you can use the above truth table to help show that not A => not D is the same as D => A
Let X₁=not A, Y₁=not D
Let X₂=D, Y₂=A

     |         not A=>not D  |           D=>A
A D  | X1 Y1       X1=>Y1     | X2 Y2    X2=>Y2
=====|=======================|=================
0 0  |  1 1         1        |  0 0       1   
0 1  |  1 0         0        |  1 0       0   
1 0  |  0 1         1        |  0 1       1   
1 1  |  0 0         1        |  1 1       1

No, this is new concept for me . thank you.

 

[Dr.Peterson]

I'll try looking for one that seems like the right level, or perhaps others will.

You probably don't want a whole book on logic, or anything too deep. I think something like this or this (chapters on logic from free courses) may be what you need. It looks like each covers everything you need (including things like truth tables, contrapositives, quantifiers).

 

[Qwertyuiop[]]

You probably don't want a whole book on logic, or anything too deep. I think something like this (a chapter from a free course) may be what you need. It looks like it covers everything you need (including things like truth tables, contrapositives, quantifiers).

thank you very much for investing your time in this. very grateful. Will take a look and do some practice .

 

[JeffM]

"Frankly. I have no clue as to what course you are doing?"
I am preparing for a university entrance test for a mechanical engineering degree. And I believe these are some questions that appeared in the test in previous sessions. I really had no clue how to go about solving these so I came and posted here. Are there any resources i can find online to help me prepare for the test? Down below I have attached the "Guide to the test" I received from the uni and this pic outlines the topics you can get in Logical reasoning questions. I have no idea what are those. Also is there a more detailed guide on the property/properties you quoted in your answer that i can find on the web? thanks

As I said in your previous question, it frequently helps to use symbols. PKA has suggested using the formal symbolism of propositional calculus and truth tables for all such problems. Here is a cheat sheet.

Logic Notes

www.cs.trincoll.edu

I suggest that you read that until you are confident that you fully understand what it means. The problem with this cheat sheet is that it does not help with quantifiers. To get a basic understanding of those, look here:

LPT Propositional Functions and Quantifiers

Again study it unril it all makes sense.

Now I am not as confident as PKA that applying propositional calculus without training is as easy as he may think. Bur in this problem, you have affirmative and negative conditionals.

Do as PKA suggests and show what conditionals you know along with their negative equivalents.

[math]C \implies D \equiv \neg D \implies \neg C\\ \neg A \implies \neg D \equiv D \implies A \\ A \implies B \equiv \neg B \implies \neg A [/math]
So you have six chains that you are to consider true

Your first problem starts with A. You have only one chain that starts with A, and it ends with B. Do you have any chains that start with B. No. So you cannot deduce anything about C from A.

Your second problem says no one will go. You have chains that end with not A, not C, and not D. You do not have a chain that ends with not B. So you cannot deduce that Barbara will not go so you cannot deduce that all will not go.

Your third problem says they all go. You have no chain that ends with C. So you cannot deduce C, which means you cannot deduce all will go.

Your fourth problem says if B is negative, then C is negative. We have only one chain that starts with [imath]\neg[/imath]B so we try starting there. That leads to [imath]\neg[/imath]A. We have only one chain that starts with [imath]\neg[/imath]A, and it leads to [imath]\neg[/imath]D. We have only one chain that starts with [math]\neg[/math]D.

So here is a chain we can build

[math]\neg B \implies \neg A \implies \neg N \implies \neg C[/math]
We can deduce that if B does not go, then C will not go.

This is very much brute force, and PKA can likely show you more elegant ways to do this exercise. But it does show how you can use the propositional calculus for such problems.



Your second problem

 

[Qwertyuiop[]]

As I said in your previous question, it frequently helps to use symbols. PKA has suggested using the formal symbolism of propositional calculus and truth tables for all such problems. Here is a cheat sheet.

Logic Notes

www.cs.trincoll.edu

I suggest that you read that until you are confident that you fully understand what it means. The problem with this cheat sheet is that it does not help with quantifiers. To get a basic understanding of those, look here:

LPT Propositional Functions and Quantifiers

Again study it unril it all makes sense.

Now I am not as confident as PKA that applying propositional calculus without training is as easy as he may think. Bur in this problem, you have affirmative and negative conditionals.

Do as PKA suggests and show what conditionals you know along with their negative equivalents.

[math]C \implies D \equiv \neg D \implies \neg C\\ \neg A \implies \neg D \equiv D \implies A \\ A \implies B \equiv \neg B \implies \neg A [/math]
So you have six chains that you are to consider true

Your first problem starts with A. You have only one chain that starts with A, and it ends with B. Do you have any chains that start with B. No. So you cannot deduce anything about C from A.

Your second problem says no one will go. You have chains that end with not A, not C, and not D. You do not have a chain that ends with not B. So you cannot deduce that Barbara will not go so you cannot deduce that all will not go.

Your third problem says they all go. You have no chain that ends with C. So you cannot deduce C, which means you cannot deduce all will go.

Your fourth problem says if B is negative, then C is negative. We have only one chain that starts with [imath]\neg[/imath]B so we try starting there. That leads to [imath]\neg[/imath]A. We have only one chain that starts with [imath]\neg[/imath]A, and it leads to [imath]\neg[/imath]D. We have only one chain that starts with [math]\neg[/math]D.

So here is a chain we can build

[math]\neg B \implies \neg A \implies \neg N \implies \neg C[/math]
We can deduce that if B does not go, then C will not go.

This is very much brute force, and PKA can likely show you more elegant ways to do this exercise. But it does show how you can use the propositional calculus for such problems.



Your second problem

That was really well explained, you made it very easy. Thank you. I like this brute method just need some practice. I have more questions that are similar to this one : D . Just one last thing: This type of If - Then question, what topic does it belong to out of these 5 topics ? Because there will be 5 questions in the logic reasoning section so I am assuming there will be one from each topic. I am trying to make a topic wise compilation of questions so when i see a question i know what type of question it is.

 

[JeffM]

Sorry. I see some typos in my post, but they did not seem to bother you so I shall ignore them.

I am not quite sure what they mean by orderings other than what was in your first thread on this topic. That involved the less than, greater than, etc relations. I am not sure that they are not most easily addressed by the symbols <, >, and so on although you can translate them into propositional calculus.

I gave you a simple cheat sheet on quantifiers.

I think the other three topics are involved in this problem.

In other words, you have already dealt with at least four of the topics in these two threads.

Problems involving quantifiers will involve words like “all,” “exists” meaning “at least one,“ etc. They will involve problems that implicitly involve quantities.