Basic predicate logic exercises

I'm working through some exercises in preparation of a test, I just have the answers, no motivation for why something is wrong. Which is essential in learning imo, so I hope you could give some pointers :)

Given:

J: John.

P: Pete.

C(x, y): x is a child of y.

M(x): x is a man.

W(x): x is a woman.

Write as a predicate:

**John or Pete has a child:**

My solution: .

The solution given is: .

Is my solution wrong? And so, why? Am I not allowed to use the scope of the quantifier like this in this context? Is my solution solution actually read as 'There's a child that belongs to either John or Pete' by using scope like this?

**Everybody has a father and mother:**

My solution:

The given solution:

Again, basically the same question: is my use of scoping correct? Should I prefer to rewrite with the quantifiers out? Is this like not simplifying a fraction?

Re: Basic predicate logic exercises

Quote:

Originally Posted by

**Lepzed** I just have the answers, no motivation for why something is wrong. Which is essential in learning imo

I agree.

Quote:

Originally Posted by

**Lepzed** Given:

J: John.

P: Pete.

C(x, y): x is a child of y.

M(x): x is a man.

W(x): x is a woman.

Write as a predicate:

**John or Pete has a child:**
My solution:

.

The solution given is:

...

Is my solution solution actually read as 'There's a child that belongs to either John or Pete' by using scope like this?

Yes. The original sentence sounds more like: John has his own child or Pete has his own. However, both statements are equivalent because

.

This is because is essentially a big disjunction , and disjunction is commutative and associative. In contrast,

because is a big conjunction, and conjunction and disjunction distribute over each other in a more complicated way.

Quote:

Originally Posted by

**Lepzed** **Everybody has a father and mother:**
My solution:

The given solution:

In the first formula, should be . Again, these formulas are equivalent because

*provided* y is not free in A. Using this fact, you can pull both existential quantifiers up front.

Re: Basic predicate logic exercises

Re: Basic predicate logic exercises

I hope you (or someone else) wouldn't mind giving this a glimpse, to see if I get this right.

**There is a real number which is greater than 0 and there is a real number which is smaller than 0**: . And not , right?

**All elements of N are greater than 10 or all elements of N are smaller than 25**: and not , right?

**There is a number in R which is greater than 10 and smaller than 25**:

is allowed though. Right?

**There is no real number which is both greater than 0 and smaller than 0**: .

Is this correct? Am I missing the point?

Re: Basic predicate logic exercises

is not a well-formed formula because ":" is not a connective. It should be . Sometimes serves a contraction for . It is also possible that or is a not a contraction, i.e., the formula syntax requires that every quantified variable has an explicit range, but I think this is less likely in your case. Anyway, double-check the formula syntax.

should be . Common idioms are and .

You are correct about putting conjunctions/disjunctions inside or outside the quantifiers.

Re: Basic predicate logic exercises

To elaborate the usage of ':', as far as I'm able to with my knowledge :p

The : is a notation used in a book we're using (Logical Reasoning: A First Course, by Nederpelt), it's derived from the set-notation, and read/used like the '|' in .

So we see predicates formulated in a sort of mix between set theory and logic, probably because the book serves as in introduction to both sports in the context of a computer science course. Like: and for example.

Taking this difference in notation in account, my formulas were correct? Goody (Nod)

Edit: Oh I see I didn't use the '[' and ']' in my solutions, (Lipssealed)

Re: Basic predicate logic exercises

Re: Basic predicate logic exercises

All right, thank you for clarifying. You've been very helpful! I'm going to practice some more and call it a day. Cheers :)