Unpacking the Mystery of the Empty Set

[cragar]

If the empty set has no elements then how can it have subsets.
Or are we just saying because they don't have anything in them there are equal.
If a set is equal to its subset then why is it a subset.
And How many subsets does the empty set have.

 

[micromass]

Hi cragar!

If a set is equal to its subset then why is it a subset.

Firstly, a subset can equal the entire set. I think that is makes sense to say that {1,2} is a part of {1,2}? Thus it makes sense to say that {1,2} is a subset of {1,2}.
Subsets that don't equal the set are called proper subsets. Don't confuse yourself with that.

If the empty set has no elements then how can it have subsets.
Or are we just saying because they don't have anything in them there are equal.
If a set is equal to its subset then why is it a subset.
And How many subsets does the empty set have.

The empty set has only one subset: the empty set itself. It must be a subset, since I said above that every set has itself as subset. It doesn't have more subsets since the empty set contains no elements.

Is there more I can clarify for you?

 

[cragar]

Why couldn't I say the empty set had an infinite amount of empty sets as subsets.

 

[micromass]

Why couldn't I say the empty set had an infinite amount of empty sets as subsets.

Because there is only one empty set. All the empty sets that you'd have as subsets are equal to each other.
Using your reasoning, we would say that {1,2} has an infinite number of {1,2} sets as subsets. But all these sets equal each other.

 

[Fredrik]

I would just like to add that the definition of ZF(C) set theory (maybe every set theory) includes an axiom that says that two sets are equal if and only if they have the same members. So if A and B have no members, we must have A=B. That's why there's only one empty set.

 

[SteveL27]

If the empty set has no elements then how can it have subsets.
Or are we just saying because they don't have anything in them there are equal.
If a set is equal to its subset then why is it a subset.
And How many subsets does the empty set have.

It might be helpful to appeal to the definition of a subset.

We say that A \subseteq B if

\forall x [x \in A \rightarrow x \in B]

In other words if x is an element of A, then x must be an element of B.

Now, let's consider the proposition \emptyset \subseteq \emptyset

Is it true that if x \in \emptyset then x \in \emptyset?

Well, yes. Because this is another one of those vacuous empty set propositions. If x is in the empty set, pretty much anything you can say about x is true. There is no x in the empty set that could falsify the left side of the implication. So the proposition "if x is an element of the empty set, then x is an element of the empty set" is vacuously true.

Therefore, by definition, \emptyset \subseteq \emptyset

For the same reason, every set is a subset of itself. Symbolically, to show that for any given set A, A \subseteq A, we just go back to the definition.

If a \in A, then a \in A. Therefore A is a subset of A.

You are right that that seems a bit strange; and in fact we give a special name to a subset that is not the entire set: we call a subset a proper subset if it's a subset that's not all of the original set. So A is a subset of A, but not a proper subset of A. And \emptyset is a subset, but not a proper subset, of \emptyset.

 

[SW VandeCarr]

Why couldn't I say the empty set had an infinite amount of empty sets as subsets.

As micromass said, the empty set has itself as a subset, but only one subset is allowed. However, the empty set can be nested within itself an infinite number of times in a cascade in which each subset contains itself as a subset. The mathematician Giuseppe Peano used this technique to define the positive integers in terms of the empty set.

3 = {0, 1, 2} = {0, {0}, {0, {0}}} = { { }, {{ }}, {{ }, {{ }}} }.

Obviously 10^{100} would take inconveniently more space.

 

[cragar]

Is it true that if x \in \emptyset then x \in \emptyset?

Well, yes. Because this is another one of those vacuous empty set propositions. If x is in the empty set, pretty much anything you can say about x is true. There is no x in the empty set that could falsify the left side of the implication. So the proposition "if x is an element of the empty set, then x is an element of the empty set" is vacuously true.

when you say that x \in \emptyset
How could this be true, the empty set has nothing in it.
But I do understand the empty set a lot better now. And thanks everyone for your replies.
And also steveL27 what do you mean when you say " it is vacuously true " I looked the word up but still not completely sure what it means.

 

[SteveL27]

when you say that x \in \emptyset
How could this be true, the empty set has nothing in it.
But I do understand the empty set a lot better now. And thanks everyone for your replies.
And also steveL27 what do you mean when you say " it is vacuously true " I looked the word up but still not completely sure what it means.

That's a good question, let me try to explain.

If x \in \emptyset then x is a purple giraffe.

This is a true statement! How can that be? Well, how could it possibly be false? In order for you to prove it false, you'd have to exhibit an element of the empty set that is NOT a purple giraffe. But you can't do that, because there IS no element of the empty set!

That's what I meant by "vacuously true." ANY statement about an element of the empty set is true ... because in order for the statement to be false, you'd have to exhibit some element of the empty set for which the statement isn't true.

So IF x is an element of the empty set, THEN x is an element of the empty set. That's a true statement ... even though we're talking about something that doesn't exist. That's why they call it vacuous.

Here is the Wikipedia definition:

A vacuous truth is a truth that is devoid of content because it asserts something about all members of a class that is empty or because it says "If A then B" when in fact A is inherently false. For example, the statement "all cell phones in the room are turned off" may be true simply because there are no cell phones in the room. In this case, the statement "all cell phones in the room are turned on" would also be considered true, and vacuously so.

The article goes on to give an extensive discussion of all aspects of the topic, including applications to the empty set.

http://en.wikipedia.org/wiki/Vacuous_truth

This kind of reasoning comes up all the time when dealing with the empty set. Once you get used to it, it seems normal.

 

[cragar]

can I say \emptyset \in \emptyset

 

[HallsofIvy]

Well, you can say it- but it is a false statement.. Being a subset is very different from being a member of a set.

In fact, the statement "x\in \phi" is false for every x.

 

[micromass]

Note the if-then statement of Steve! He said

If x\in\emptyset then x is a purple giraffe.

In an if-then statement, the if-statement can be false, and if the if-statement is false, then the conclusion can be everything. We can even have

If 1+1=1 then I am the pope.

This is a true statement! (an amuzing aside: a famous proof for that is as follows. The pope is 1, I am one, therefore I and the pope are one).

This behaviour of if-then statements is often called "ex falso sequitur quodlibet" which means: from a falsehood follows whatever you want!

However, if the if-statement is a true statement, then the then-statement must be true. So you cannot say

If 1+1=2 then I am the pope.

This is false. As you can see, if-then statements are only interesting if the if-statement is true...


I hope that clears things up...

 

[cragar]

Ya I remember the truth table for if-then statements.