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### Union and Intersection of Empty Family of Sets

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

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.

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

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/
```
Associated Topics:
High School Logic
High School Sets

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