Predicate Logic and Sets

What is the basis for reason? And mathematics?

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nguyentruongnghia
Posts: 1
Joined: Sun Sep 11, 2022 9:42 pm

Predicate Logic and Sets

Post by nguyentruongnghia »

Hi friends,

I don't seem to be able to formalize a statement about the equivalence of two sets. Specifically, how do I formalize "the number of apples in this box A is the same as the number of pears in that box B". Surely one has to use universal and existential quantifiers in some way, and I suppose Russell must have done it 100 years ago, but I can't do it, and just can't find the formula anywhere.

Thanks
Nghia
wtf
Posts: 1087
Joined: Tue Sep 08, 2015 11:36 pm

Re: Predicate Logic and Sets

Post by wtf »

nguyentruongnghia wrote: Sun Sep 11, 2022 9:56 pm Hi friends,

I don't seem to be able to formalize a statement about the equivalence of two sets. Specifically, how do I formalize "the number of apples in this box A is the same as the number of pears in that box B". Surely one has to use universal and existential quantifiers in some way, and I suppose Russell must have done it 100 years ago, but I can't do it, and just can't find the formula anywhere.

Thanks
Nghia
One way is to say that there's a bijection between the two sets. That means that there is a bijective function between them.

A function is a relation such that every element of A appears exactly once; that is, at least once and no more than once.

A relation is a subset of the Cartesian product A X B.

A bijective function is a function that is injective, meaning that no two elements of A get mapped to the same element of B; and surjective, meaning that every element of B gets hit by some element of A.

So you have to put into symbolic logic the concepts of relation, function, injection, and surjection. That's a bit of work if you have to do it from scratch.

But if you're allowed to assume that you know what a bijective function is, you can say that A and B have the same cardinality if there exists f : A -> B such that f is bijective. It's up to you how much of that you want to drill down to first principles.
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