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March 29, 2024, 01:39:16 am

Author Topic: Logic + Proof questions  (Read 2141 times)  Share 

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#1procrastinator

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Logic + Proof questions
« on: April 13, 2012, 11:31:57 am »
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I'm reading through a logic + proof book and I thought I'd start a question thread for it as there's not many solutions to the exercises

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http://en.wikipedia.org/wiki/List_of_logic_symbols


Analyse the logical forms of the following:


1) We'll have either a reading assignment or homework problems, but we won't have both homework problems and a test. Let H be 'we'll have a homework assignment', R for the reading an T for the test.

My answer:
(R ⋀ T) ⋁ (R ⋀¬T) ⋁ H
Given answer:
(R ⋁ H) ⋀ ¬(H ⋀ T)

Obviously they're different but I wonder whether the way I wrote my answer is acceptable or not (or if I overlooked something and is just wrong).

Also, is this second part ¬(H ⋀ T) in this context interpreted as if there's homework, then there's no test? IS ¬(H ⋀ T) the same as ¬H ⋀ ¬T

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2) √(7) ≰ 2

My answer: ¬((√(7) < 2) ⋀ ( √(7) = 2))

The given answer used ⋁ rather than ⋀. I realise that that's because the symbol is 'not less than OR equal to' but since we already know that it's neither less than or equal to, couldn't we use and?


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3) How would you analyze the logical form of '3 is a common divisor of 6, 9, and 15'? Would you just let a letter represent '3 is a common divisor of 6' and for the other two as well and write P ⋀ Q ⋀ R?

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4) I'm stuck on these two:
Let S stand for the statement 'Steve is happy' and G for 'George is happy'. What English sentences are represented by the following:

a) [S ⋁ (G ⋀ ¬S)] ⋁ ¬G
b) S ⋁ [G ⋀ (¬S ⋁ ¬G)]

For a)
a) Either Steve is happy or George is happy and Steve is unhappy, or George is unhappy.

So if George is NOT unhappy, meaning he IS happy, then Steve is unhappy . If he is not unhappy, can Steve be happy? Does 'Steve is happy' alone imply that George is also happy? Or can we not say anything about that?


Thanks

kamil9876

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Re: Logic + Proof questions
« Reply #1 on: April 14, 2012, 01:16:42 am »
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I'm reading through a logic + proof book and I thought I'd start a question thread for it as there's not many solutions to the exercises

----

http://en.wikipedia.org/wiki/List_of_logic_symbols


Analyse the logical forms of the following:


1) We'll have either a reading assignment or homework problems, but we won't have both homework problems and a test. Let H be 'we'll have a homework assignment', R for the reading an T for the test.

My answer:
(R ⋀ T) ⋁ (R ⋀¬T) ⋁ H
Given answer:
(R ⋁ H) ⋀ ¬(H ⋀ T)

Obviously they're different but I wonder whether the way I wrote my answer is acceptable or not (or if I overlooked something and is just wrong).


It's wrong, your one allows the scenario of having all three things, i.e: if R,H,T are all true then (R ⋀ T) ⋁ (R ⋀¬T) ⋁ H  is true, simply because H is.

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Also, is this second part ¬(H ⋀ T) in this context interpreted as if there's homework, then there's no test?


Break it down into pieces: (H ⋀ T) means "there's homework and there's a test" so ¬(H ⋀ T) means "it is not true that there is both homework and a test".

"if A then B" is by definition "¬A ⋀ B" which is not the same as ¬(A ⋀ B)


Quote
IS ¬(H ⋀ T) the same as ¬H ⋀ ¬T

No, you have to change or with and when expanding out the negation, i.e ¬(H ⋀ T) is actually ¬H⋁¬T  (see De Morgan's laws)

Quote
The given answer used ⋁ rather than ⋀. I realise that that's because the symbol is 'not less than OR equal to' but since we already know that it's neither less than or equal to, couldn't we use and?

I presume the given answer also doesn't have that negation out at the front. Both answers are the same again by De Morgan's laws.

Quote
3) How would you analyze the logical form of '3 is a common divisor of 6, 9, and 15'? Would you just let a letter represent '3 is a common divisor of 6' and for the other two as well and write P ⋀ Q ⋀ R?

Unfortunately in propositional logic that's probably the best you could do. Which is why predicate logic is more interesting.





« Last Edit: April 14, 2012, 01:20:38 am by kamil9876 »
Voltaire: "There is an astonishing imagination even in the science of mathematics ... We repeat, there is far more imagination in the head of Archimedes than in that of Homer."

#1procrastinator

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Re: Logic + Proof questions
« Reply #2 on: April 24, 2012, 05:53:34 pm »
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^ Thanks Kamil...have to put the logic/proof reading on hold (again) for the moment cause of VCE stuff and constant procrastination ><