Union and Intersection of Empty Family of Sets

Date: 03/20/2003 at 03:28:17
From: Jonathan
Subject: Set theory

I have not found a satisfactory proof of the statements:

 i) the union of an empty collection of sets is the empty set; and
ii) the intersection of the empty collection of sets is the universal 
    set.

Can anyone provide me proofs of the above?

An empty collection of sets is no sets, so how can a union or 
intersection be formed at all? The concept that the intersection of an 
empty set can produce something other than another empty set is 
confusing.

Date: 03/20/2003 at 07:51:58
From: Doctor Jacques
Subject: Re: Set theory

Hi Jonathan,

The question is really: In what sensible way can we define the union 
and intersection of an empty family of sets?

From an intuitive point of view, we can note that, if the family is 
enlarged, the union can only get larger, and the intersection can 
only get smaller. This suggests that, for an empty family, the union 
is the smallest possible set, and the intersection is the largest 
possible set.

We can also consider this from a more formal point of view. A family 
of sets can be described as:

  {A_i | i in I}

where I is an indexing set (not necessarily finite or even countable).

The union of the family is the set of elements x such that:

  there exists an i in I such that x is in A_i

and this is always false if I is the empty set, because "there exists 
something in I" is always false. As no element x statisfies the 
definition, the union is the empty set.

The intersection of the family is the set of elements x such that:

 for every i in I, x is in A_i

and this is always true if I is empty, because there is nothing to 
check (this may seem strange, see below). As any element x makes the 
statement true (x is not even used), the intersection is the set of 
all elements in the given context - i.e. the "universal set."

More generally, if A happens to be the empty set, any statement of 
the form

  "there exists a in A such that (something)"

becomes false, since it means:

  "there exists x such that x is in A and (something)"

In this case, the first proposition (x is in A) is false, and the 
main connective is AND, so the whole proposition is false.

In the same way, if A is the empty set, any statement of the form

  "for all x in A (something)"

is true, since it means:

  "for all x, x is in A --> (something)"

This is of the form P --> Q, where P is "x is in A" and is always 
false. As you should remember, P --> Q means "P is false or Q is 
true." In this case, P is false, and P --> Q is therefore true.

Does this help?  Write back if you'd like to talk about this some 
more, or if you have any other questions.

- Doctor Jacques, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 03/21/2003 at 11:51:23
From: Jonathan
Subject: Set theory

Hello Dr. Jacques,

Thank you for your reply. I understand the first section of the 
explanation up to where you begin to talk about the propositions, in 
particular the part about the intersection (P --> Q, etc). I'm not 
sure I follow this bit too well.

Is this sort of explanation described as logic? I ask this because I 
have a book about set theory and logic which I can go through if you 
think this would help me. Incidentally, the section in this book 
about the empty collection is not too good - I had tried this 
already before I asked my question!

Regards,
Jonathan

Date: 03/22/2003 at 07:01:25
From: Doctor Jacques
Subject: Re: Set theory

Yes, this is indeed formal logic, and you should definitely study the 
subject, at least up to some point.

The aim of formal logic is to describe rules that allow you to decide 
whether or not a particular argument is valid, using a mechanical 
procedure (i.e. without even knowing what it is all about...). There 
are computer programs that can verify the correctness of a proof (of 
course, the proof must be stated in a language that computers can 
process).

Formal logic is required as a foundation of mathematics, to ensure 
that the arguments used are not subjective.

Starting at the second half of the 19th century, there was an effort  
to establish the foundations of mathematics on strict formal logic 
rules.

To get an idea of the methods involved, you might look at:

   Introduction to Symbolic Logic - Dr. Achilles, Dr. Math FAQ
   http://mathforum.org/dr.math/faq/symbolic_logic.html 

- Doctor Jacques, The Math Forum
  http://mathforum.org/dr.math/