# Infinite collections???

• May 11th 2010, 02:39 PM
dch
Infinite collections???
Does there exist an infinite collection of sets such that the intersection of every two sets in the collection is nonempty, but the intersection of every three distinct sets in the collection is empty???

Could you also explain what sets are???

Thanks
• May 12th 2010, 12:54 AM
Swlabr
Quote:

Originally Posted by dch
Could you also explain what sets are???

A very good question, and one I suspect people don't ask often enough! Unfortunately, the answer is overly complicated (it is based on a set of axioms which are not nice at all, called `ZFC'). However, wikipedia seems to cover the topic well, so I would recommend reading that page then trying your question. If you still can't crack it, ask it again!

Anyway, a basic notion of sets is a collection of objects, called elements. These elements are normally numbers (because this is maths). To show that the collection of elements is a set we enclose it in curly brackets, for example $S=\{1, 2, 3, 4, 5\}$ is a set. An exampls of an element of $S$ would be 3, and 4 would be another one. 6, however, is not an element.

Sets cannot have more than one copy of an element. So, for example, $\{1,2,1\} = \{1,2\}$.

Now, there are a few operations defined on sets. The main ones are union and intersection.

$\{A\} \cup \{B\} = \{A, B\}$ then cancel any multiple elements. So, for example, $\{1, 2, 3, 4, 5\} \cup \{1, 4, 6, 7\} = \{1, 2, 3, 4, 5, 1, 4, 6, 7\} = \{1, 2, 3, 4, 5, 6, 7\}$.

Intersection gives you the subset of elements contained in both of the sets. So, for example, $\{1, 2, 3, 4, 5\} \cap \{1, 4, 6, 7\} = \{1, 4\}$ as the elements 1 and 4 are the only elements in both of the sets.

Does that make sense?
• May 12th 2010, 07:40 AM
MoeBlee
Quote:

Originally Posted by Swlabr
elements are normally numbers (because this is maths).

And mathematics includes set theory (and, in a certain sense, vice versa). There is no requirement that elements normally be numbers. Ordinary mathematics (which includes such subjects as analysis, topology, abstract algebra, et. al) has many kinds of objects that are not numbers, and such objects (if they are not proper classes) are themselves elements of certain sets.
• May 12th 2010, 07:53 AM
MoeBlee
Quote:

Originally Posted by dch
Could you also explain what sets are???

There are at least four different answers:

(1) Naive, intuitive, everyday mathematical sense of the word 'set': Sets are collections (or "classes") of objects. The objects collected into a set are the elements of the set.

(2) The notion of 'set' is taken as an undefined basic concept.

(3) In certain formal theories, we may define the predicate 'is a set' in certain ways. For example, one may find in a theory such as Z set theory (without urelements) that a reasonable definition of 'is a set' is:

x is a set if and only if x is an element.

The purpose of that definition is to distinguish sets from proper classes. That is, a set is an object that is a member of some class, while a proper class is not a member any class.

In such a theory, all objects are sets, since every object x is a member of {x}.

(4) In certain formal theories, we take the predicate 'is a set' as a primitive predicate. For example, in Bernays class theory, 'is a set' is primitive.

/

Generally, we might distinguish objects in this way:

urelement
class
set
proper class

An urelement is an object that has no members (except for the empty set which is the only non-urelement having no members).

A class is an object that is either the empty set or has members.

A set is a class that is a member of some class.

A proper class is a class that is not a member of any class.
• May 12th 2010, 09:31 AM
Swlabr
Quote:

Originally Posted by MoeBlee
And mathematics includes set theory (and, in a certain sense, vice versa). There is no requirement that elements normally be numbers. Ordinary mathematics (which includes such subjects as analysis, topology, abstract algebra, et. al) has many kinds of objects that are not numbers, and such objects (if they are not proper classes) are themselves elements of certain sets.

Well, sure, but I can never quite understand why lecturers always have contrived examples such as {cat, dog, house}. Why don't they just start with numbers, then progress to, for example, permutations or symmetries or whatever. Just using numbers is sufficient for a first course.
• May 12th 2010, 09:45 AM
MoeBlee
Quote:

Originally Posted by Swlabr
Well, sure, but I can never quite understand why lecturers always have contrived examples such as {cat, dog, house}. Why don't they just start with numbers, then progress to, for example, permutations or symmetries or whatever. Just using numbers is sufficient for a first course.

I basically agree with you in that sense. But I guess they use concrete objects as examples in order to get across the more general, naive, intuitive, everyday sense of the word 'set'. But, as you've alluded, in ordinary abstract set theory (without urelements) the objects are all abstract, and for any finite example (even in predicate logic), using natural numbers (and sets built from them) is sufficient (any countable model can just as well be emulated with natural numbers). (Of course, even more rigorous is to refrain from using natural numbers until they have themselves been constructed in the the theory).