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## #1 2013-05-20 02:41:47

rafa_mota04
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### Mathematical Logic and Structures, URGENT!

Hey guys, i need a HUGE favor, i need the resolution for this 5 questions, its a question of end this year my degree or stay one year just with a subject. Please I am begging, who knows the resolution please say me something.

Part 1: Mathematical Logic
2-    Convert the formula into the prenex form: (square root of)x p(x,g) -> ( inverted E )y (square root of) z q (x,y)
3-    Consider the language L with the symbol of equality and one array predicate symbol. Write the formula in this language which express “there exist exactly one x such that p(x) holds”
4-    Verify, which any of the formulas ¬ A v B and A or B substituted for x makes the formula ( A -> x ) -> ( x -> B ) a tautology.
Part 2: Mathematical Structures
1 - find the subgroup of the group (Z, +, 0) generated by the set {20,25} .
2 - Describe the free object over a given set X on the constant Pos (of partially ordened sets and isotone maps)

## #2 2013-05-20 03:28:55

SteveB
Power Member

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### Re: Mathematical Logic and Structures, URGENT!

#### rafa_mota04 wrote:

Hey guys, i need a HUGE favor, i need the resolution for this 5 questions, its a question of end this year my degree or stay one year just with a subject. Please I am begging, who knows the resolution please say me something.

Part 1: Mathematical Logic
2-    Convert the formula into the prenex form: (square root of)x p(x,g) -> ( inverted E )y (square root of) z q (x,y)

I am not the world's expert on logic, but I am sometimes good at these sort of questions.

I apologise if I have got completely the wrong end of the stick with this, but I think you mean:

Not sure what exactly is required by "convert the formula into prenex form". (Is this "prenex normal form" ?)
If it is prenex normal form it may be a good idea to look at the Wikipedia info:
http://en.wikipedia.org/wiki/Prenex_normal_form
To me "prenex form" means 'there exists' (inverted E ?), 'for all' (looks like an A written vertically inverted) etc.
also includes NOT, AND, OR, IMPLIES etc. in symbolic form.
I included the "such that" bits to help me understand it, you have not included any because it
is conventional to leave them out in prenex form I suppose. Or have I completely got the wrong
understanding of the meaning of that question?
What else is there to be done ?
I am obviously missing something here.
I think it is a bit mean asking students to convert something like that and memorise all
the rules contained in that link for an exam. Are you revising for an exam ?
How can something depend on the square root of z if z is not included in the predicate ?

Clarification in LaTex:

Is p(x,g) a predicate that does a test upon the variables x and g and gives a "true" or "false" ?
Is q(x,y) similarly defined for x and y giving a "true" or "false" ?

Another question: Do you know about how to convert an implication into a logical statement about

in terms of A, B and NOT, AND, OR etc ?

So if A implies B then we mean that if A is true then B must be true.
If A is false then nothing is being stated about B. So B is either true or false for the implication
to still be acceptable.
(A implies B) is considered true if and only if ((A and B are true) OR (A is false))
If A is true and B is false then (A implies B) is considered false.
So I think it results in: (B) OR (NOT A) as the output of the predicate P(A implies B)

(If you are revising for an exam then for goodness sake do some practice questions on this
and whatever else you are supposed to learn.)
(I only know a bit of degree style logic so please double check me on the things in this post

Last edited by SteveB (2013-05-20 06:38:35)

## #3 2013-05-20 07:54:39

SteveB
Power Member

Online

### Re: Mathematical Logic and Structures, URGENT!

#### rafa_mota04 wrote:

4-    Verify, which any of the formulas ¬ A v B and A or B substituted for x makes the formula ( A -> x ) -> ( x -> B ) a tautology.
Part 2: Mathematical Structures

Let us first substitute ((NOT A) OR B) which I hope means the same as ¬ A v B into x.

Right so that makes: (A -> (¬ A v B)) -> ((¬ A v B) -> B)

(LEFT HAND SIDE PART:This is true if and only if (A is false) OR ((¬ A v B) is true).)
(RIGHT HAND SIDE PART:This is true if and only if ((¬ A v B) is false) OR (B is true).)
There are four possibilities of this. I would usually use a table for this, but this is tricky with text only.
So. Case 1. A and B are true. Since B is true (¬ A v B) is true because of the OR logic. By the implication
statement logic the whole thing is true. Now let's look at the (x -> B) bit. (¬ A v B) -> B
well B is true so the whole thing is true (true -> true) gives a true output.
Case 2: A is true, but B is false. LHS: By OR logic if both sides are false the whole thing is false.
Since (true -> false) this outputs false. RHS: The (x -> B)  ((¬ A v B) -> B) The left hand side is false so the implication
will output true. Overal: The output is true.
Case 3: A is false, but B is true. LHS: The statement simplifies to true. RHS: The statement simplifies to true. Overal: true.
Case 4: A is false, B is false. LHS: true RHS: true. Overal: true

Unless I have gone wrong there that is a tautology. I am not sure that this is the intended method and have no idea how
a course designer of something like this would intend you to present an answer to something like this. However I hope
that helps with understanding a bit. Apologies for any errors.

For (A OR B) a shortcut might be to observe that to try to produce a false output we need B to be false.
So let B be false in the expression:
(A -> (A OR B)) -> ((A OR B) -> B)
Consider the case A is true. (A OR B) is true. A is true, and B is false.
The left hand side is true (true -> true).
The right hand side is false (true -> false).
Overal: (true -> true) -> (true -> false) = (true -> false) = false
Hence this is not a tautology.

Last edited by SteveB (2013-05-21 00:28:36)

## #4 2013-05-21 02:28:31

rafa_mota04
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### Re: Mathematical Logic and Structures, URGENT!

tHANK YOU GUYS, SOMEONE KNOW: 2 - Describe the free object over a given set X on the constant Pos (of partially ordened sets and isotone maps)