Can someone confirm these English to predicate logic problems for me?

[Ownaginatious]

Homework Statement

I actually have to problems I would just like someone to confirm for me. I have several other problems similar to these, and I don't want to waste time in case I do not understand the fundamentals. Anyway:

a) Define suitable predicates and functions and then formalize the following sentence : A student receives a grade for every course in which he or she registers:

b) Translate the following English statements into predicate logic : Adding two odd integers yields an even number. Use only addition and multiplication; do not use division, mod, or predicates even(x) and odd(x).

2. Attempt at a solution

Now here is my attempt:

a) I defined the variables:
- c = courses
- s = students

And the functions:
- isReg(s, c)
- getGrades(s, c)

And then combined them into the following formal statement:

(\forall s|: (\forall c | isReg(s,c) : getsGrade(s, c)))

b) Now for this I simply derived the following:

(\forall x,y:Z|(\exists n:Z|:(2n+1) = x) \wedge (\exists n:Z|:(2n+1) = y):(\exists n:Z|:(2n) = (x + y)))

If I've gone wrong, any help at getting back onto the right track would be much appreciated.

Thanks :)

 

[sylas]

(a) is probably right... but English can be ambiguous. When the sentence says "a student..." it could also be taken to mean that "there is a student..." which would be a quite different meaning.

But I suspect you've given what was intended.

I wonder a bit about your syntax for the restricted quantifiers (using "|" to restrict the range of a variable) in part a, where you have the "|" without the preceding domain. Is there a clearly defined syntax you are using?

 

[Ownaginatious]

Ya, I was wondering that exact same thing sylas. My textbook talks a lot about how English is so ambiguous, so I'm going to assume that whether this applies to one student or all students is open to my interpretation ;).

 

[Ownaginatious]

I wonder a bit about your syntax for the restricted quantifiers (using "|" to restrict the range of a variable) in part a, where you have the "|" without the preceding domain. Is there a clearly defined syntax you are using?

Oh I was led to believe that that's the equivalent to saying "over the whole domain", which is the equivalent to just putting "true" there.

 

[sylas]

Oh I was led to believe that that's the equivalent to saying "over the whole domain", which is the equivalent to just putting "true" there.

It depends on what rules you've been given for syntax. Your syntax uses ":" for two different roles, which is ok. I've seen it done before. I'll use "•" for the start of the statement to which the quantified applies, just for clarity. Here's a possible grammar:

    statement ::= quantifier variable ":" range "•" statement
    range ::= setvariable
            | setvariable "|" statement

Basically, a grammar like this requires a range for a variable, to be either just named set, or perhaps a named set with an additional constraint or restriction. That's how I have usually seen "|" used. It can be read "such that".

The "•" can be replaced with another ":" without ambiguity, unless the first ":" becomes optional. If the first ":" is optional you might have the following rule as well:

    statement ::= quantifier variable "•" statement

In this case there has to be some understood domain that is being used.

What I have not seen is "|" by itself, without the first ":". It makes sense, if understood has having a ":X" implied before hand, where X is the understood domain.

I'm just asking in case... have you been given a syntax you are expected to use? What you have is still quite comprehensible, but it you have a required syntax you might want to check.

Cheers -- sylas

 

[Ownaginatious]

Well, as given in our book, the syntax they want us to use is the following:

(*x | R : P)

R = Range, P = Body, * = operator/function, x = variable(s). The ':' is also used to represent type (i.e. x : Z = x of type integer).

In class my professor just left out R or made it "true" when he wanted to use the whole domain.

Here is an example he did:

"Some integer is larger than 23" formally is (\exists x:Z|:x > 23) or (\exists x:Z|true:x > 23)

Sorry if I've misunderstood your question. This is all new to me.

 

[sylas]

Well, as given in our book, the syntax they want us to use is the following:

(*x | R : P)

R = Range, P = Body, * = operator/function, x = variable(s). The ':' is also used to represent type (i.e. x : Z = x of type integer).

In class my professor just left out R or made it "true" when he wanted to use the whole domain.

Here is an example he did:

"Some integer is larger than 23" formally is (\exists x:Z|:x > 23) or (\exists x:Z|true:x > 23)

Sorry if I've misunderstood your question. This is all new to me.

You've understood fine, and your answers seem fine and consistent with the syntax you've provided. I just wondered about the syntax, and you've explained it well.

Cheers -- sylas

 

[Ownaginatious]

You've understood fine, and your answers seem fine and consistent with the syntax you've provided. I just wondered about the syntax, and you've explained it well.

Cheers -- sylas

Oh awesome :) Thank you very much for you help!