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### Is an Empty Set a Subset?

Date: 08/31/2001 at 12:01:58
From: anabelle
Subject: Proof

Hello Dr. Math,

The empty set is a subset of all sets, right? What is the proof of
the example:

For any event W in the sample space S, what is the proof that the
empty set is a subset of W?

Thank you.

Date: 08/31/2001 at 12:58:04
From: Doctor Peterson
Subject: Re: Proof

Hi, Anabelle.

A subset of a given set is simply any set, all of whose elements are
contained in the other. Since the empty set has no elements, all of
its elements are in any other set! It sounds weird, but that's the way
logic works.

To put it another way, a set A is NOT a subset of B if there is some
element x of A that is not in B. Since the empty set has no elements
that are not in your given set, we can't say it is NOT a subset. That
means that it is.

To select a subset, we must look at each member of the set and decide
whether to keep it. If we say "yes" to every member, we have the set
itself; if we say "no" to all of them, we have the empty set. We could
choose to exclude these from the definition of subset, but it makes a
lot of things easier if we include them. That way there are no special
cases to deal with when we state theorems.

Here is an answer from our archives that deals with this question:

Empty Sets
http://mathforum.org/dr.math/problems/hood8.15.97.html

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

Associated Topics:
High School Sets

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