Empty Set Problem

[Ender]

Is it true that all the elements of the empty set are aliens from outer space?

[Sidoh]

The empty set is a proper subset of every other set... except... itself?

[iago]

Well, that circle with a slash through it could be a drawing of an alien, in ancient Greece.

[Camel]

Is it true that all the elements of the empty set are aliens from outer space?

Depends on how many dimensions are involved.

[Ender]

The empty set is a proper subset of every other set... except... itself?

The empty set is a subset of itself. The # subsets of a set with n elements is 2^n (which can be seen by a binary encoding). Thus the # subsets of empty set are 2^0 = 1. You can see that this 1 subset is the empty set since every element of the empty set is contained within the empty set.

EDIT: But to answer your question more technically, the empty set is not a PROPER subset of itself, since {} = {}, though it is a subset of itself.

All this discussion about subsets and proper subsets has little to do with the problem, though. Subsets are not elements of their supersets; it's just that the elements of subsets are elements of their supersets.

[Sidoh]

The empty set is a subset of itself. The # subsets of a set with n elements is 2^n (which can be seen by a binary encoding). Thus the # subsets of empty set are 2^0 = 1. You can see that this 1 subset is the empty set since every element of the empty set is contained within the empty set.

EDIT: But to answer your question more technically, the empty set is not a PROPER subset of itself, since {} = {}, though it is a subset of itself.

In case you didn't know, that's exactly what I was getting at.  I wasn't really asking a question.  It was supposed to be in the form "<Answer>... am I right?"

All this discussion about subsets and proper subsets has little to do with the problem, though. Subsets are not elements of their supersets; it's just that the elements of subsets are elements of their supersets.

How does it have little to do with it?  The answer is implied by saying that.  If the empty set is a subset of the set of space aliens, then all of the elements in the empty set are space aliens (and, in the case that there are no space aliens, the empty set is still a subset of the set of space aliens, which is equivalent to the empty set).

[Ender]

Interesting way to think about it, using the fact that the empty set is a subset of every set.

But you can also say the empty set is a subset of the set of all things that are not space aliens.

=\

[Sidoh]

There are no elements in the empty set, therefore you can say anything you want about "all of its elements."

But by that logic, the same "contradiction" arises.  All of the elements in the empty set are space aliens, but they are also not space aliens.

Is there a cleaner solution that you're looking for?

[Ender]

There are no elements in the empty set, therefore you can say anything you want about "all of its elements."

Correct! 

But by that logic, the same "contradiction" arises.  All of the elements in the empty set are space aliens, but they are also not space aliens.

It's only a contradiction if you find an element of the empty set

Is there a cleaner solution that you're looking for?

Nope.


This is why Cantor called the empty set the set of all contradictions =) And I should note that conversely, if you can find a set that has such contradictions, it must be the empty set. The empty set is cool, and it's a very seminal discovery in mathematics; there's even debate about whether it exists.

[Camel]

Every element of the empty set is not an element of the empty set. It's the set equivalent of an asymptote.

[Ender]

Every element of the empty set is not an element of the empty set.

Yes

It's the set equivalent of an asymptote.

Explain. BTW, a function is defined as a set, so no need for "set equivalent".

[Camel]

A function is a relationship between its input values and its output value; it is distinctly different from a set. You may be thinking of a map, which a function can be expressed as, although they are not equivalent, because while any function can be expressed as a map, a map is often less specific than the function that describes it. You can express the domain (input values for which there is an output value) or range (output values for which there are input values) of a function as a set (and that's generally how it's done), but those two things do not uniquely describe the function either.

One example of an asymptote is ; it approaches and where and but never reaches it. One could reasonably argue that . If you try to approximate by finding the , you get ; if you try to approximate by finding the you get .

Does that make more sense?

[edit] tex

[Ender]

A function is a relationship between its input values and its output value; it is distinctly different from a set. You may be thinking of a map, which a function can be expressed as, although they are not equivalent, because while any function can be expressed as a map, a map is often less specific than the function that describes it. You can express the domain (input values for which there is an output value) or range (output values for which there are input values) of a function as a set (and that's generally how it's done), but those two things do not uniquely describe the function either.

One example of an asymptote is ; it approaches and where and but never reaches it. One could reasonably argue that . If you try to approximate by finding the , you get ; if you try to approximate by finding the you get .

Does that make more sense?

[edit] tex

No, a function is defined as a set. A function f : A -> B is defined as a subset of A x B (where the set A x B is the direct product of A and B) s.t. for each a in A there exists a unique b in B with (a, b) in f.

EDIT: Pretty much everything in math is derived from sets (hence Set Theory), although Category Theory messes things up a bit.

[Camel]

No, a function is defined as a set. A function f : A -> B is defined as a subset of A x B (where the set A x B is the direct product of A and B) s.t. for each a in A there exists a unique b in B with (a, b) in f.

EDIT: Pretty much everything in math is derived from sets (hence Set Theory), although Category Theory messes things up a bit.

The set described by a function does describe the function, but it does not identify it! In set theory, it is not a goal to return to the original function, although it's almost always trivial to do so, because of the simple fact that distinct functions with equivalent input-output are distinct only in their definition, and not in their behavior.

Set theory is nothing more than a mental aid for mathematics; it isn't required for deriving anything. Because it's based simpler principles, it's possible to bypass it. Doing so would be equivalent to reducing an integral to a summation, or a derivative to a limit.

[Ender]

One example of an asymptote is ; it approaches and where and but never reaches it. One could reasonably argue that . If you try to approximate by finding the , you get ; if you try to approximate by finding the you get .

Does that make more sense?

[edit] tex

As an informal argument, it's a sensible analogy. But it's not the asymptote that's the empty set, it's f(0), where f(x) = 1/x : R -> R.

[Ender]

No, a function is defined as a set. A function f : A -> B is defined as a subset of A x B (where the set A x B is the direct product of A and B) s.t. for each a in A there exists a unique b in B with (a, b) in f.

EDIT: Pretty much everything in math is derived from sets (hence Set Theory), although Category Theory messes things up a bit.

The set described by a function does describe the function, but it does not identify it! In set theory, it is not a goal to return to the original function, although it's almost always trivial to do so, because of the simple fact that distinct functions with equivalent input-output are distinct only in their definition, and not in their behavior.

Set theory is nothing more than a mental aid for mathematics; it isn't required for deriving anything. Because it's based simpler principles, it's possible to bypass it. Doing so would be equivalent to reducing an integral to a summation, or a derivative to a limit.

No, you're wrong. f is defined as a set of ordered pairs, viz. {(a, b), (c, d),...}. The inverse function is a set of the same form. Properties of functions like surjectivity, injectivity, and bijectivity are easy to define with sets. The natural numbers are constructed with sets, and the reals too, although there's more to it, like Dedekind cuts.

[Camel]

Okay, so I actually did some research in to this to settle the topic.

The definition of a function is the rule that describes the relationship between the elements of its range and the elements of its domain. The tricky part is that there's controversy over how the word "rule" should be defined. One interpretation of the word rule is what you said: the binary relation its self. The other school of thought says that the binary relation must come from the function, otherwise it's just an arbitrary set with no definition.

Technically, we're both correct under different definitions of the word "rule," although this is totally irrelevant in the context of the thread, so back to the topic:

The asymptote is the representation of the thing that is missing!

Also, it doesn't make much sense to describe a subset of an empty set, lol.