Understanding Sets within Sets: Empty Set & Subsets

[Solid Snake]

Ok. I don't understand sets within sets. No one has been able to explain it to me simply.

So you can have sets inside of sets. That I get. But what happens when you get to something tricky like the empty set? For example, I know that ø ε {ø} is true. I don't understand how. How do I read this? Does this mean that "the empty set is in a set that contains the empty set"?

How do I read "{{ø}} is a subset of {ø, {ø}}"? I think I read it as "a set that contains a set that contains the empty set is a subset of a set with the elements of the empty set, and a set that contains the empty set"? Is this how to read it? Thanks.

 

[Simon Bridge]

You mean $\emptyset \in \{\emptyset \}$ ??

Like $a\in\{ a\}$ says that the letter "a" is a member of the set that contains only the letter "a".
also $\{a\}\subset \{a,b,c\}$ etc. Something like $\{\{a\}\}$ is just a lot of redundant brackets.

When a set only has one member, a lot of the notation becomes redundant.

 

[gopher_p]

Maybe think of sets like boxes, and the elements are the contents of the box. Two boxes are considered to be the "same" if their contents are the same. The empty set is like a box with nothing in it.

So $\{\emptyset\}$ is a box containing an empty box, $\{\{\emptyset\}\}$ a box that contains a box which contains an empty box (kind of like nesting dolls), and $\{\emptyset,\{\emptyset\}\}$ is a box that contains two boxes, one of which is empty and one of which contains an empty box.

If you extend the analogy to other sorts of containers and also consider things like buildings and semi trailers to be "boxes", you'll find all sorts of tangible real-life examples. Of boxes inside other boxes. The analogy isn't completely apt, though. You'll have to decide for yourself how you want to treat boxes that have multiple kinds of the "same" object.

 

[mal4mac]

Ok. I don't understand sets within sets. No one has been able to explain it to me simply.

So you can have sets inside of sets. That I get. But what happens when you get to something tricky like the empty set? For example, I know that ø ε {ø} is true. I don't understand how. How do I read this? Does this mean that "the empty set is in a set that contains the empty set"?

"The empty set is in a set that contains the empty set" is correct!

I'd be tempted to put the ε upfront and read it more directly as "the empty set "is an element of" the set that contains the empty set".

As you've got that, do you really need to bother translating {ø, {ø}} into English? Part of the reason we use math is because it's more succinct than English sentences, and saves a lot brain strain. If I say "This set has two elements, one of which is the empty set", would you agree? Do you see it? If so you've probably got it!

 

[micromass]

Something like $\{\{a\}\}$ is just a lot of redundant brackets.

How are the brackets redundant??

 

[Stephen Tashi]

Ok. I don't understand sets within sets. No one has been able to explain it to me simply.
[
So you can have sets inside of sets. That I get.

You aren't asking a specific question because you are using words like "within" and "inside" that don't have a technical definition in set theory instead of using the standard mathematical relations "is a member of" and "is a subset of".

I know that ø ε {ø} is true.

It's only true or false if it has a definite meaning. What do you mean by that notation?

 

[FactChecker]

I know that ø ε {ø} is true. I don't understand how.

You need to be careful here. Any set contains the empty set as a subset, but not as an element. When you write A = {ø}, then you are saying that ø is an element of A. We take your word for it. So ø ε {ø} only because you put it in there. And ø \not\in{a} unless you say a = ø.

 

[Stephen Tashi]

Any set A defined in some context has the property that the null set defined in that context is a subset of A. However, saying that a null set defined in one context is a subset of a set defined in a different context may be a true statement, a false statemen, or it may have no meaning - i.e. it may not be a statement at all. For example if one context is sets of real numbers and another context is sets of people, then it is not meaningful to say "The null set of people is a subset of the set of rational numbers". To give such a statement meaning, you must establish a context where a statement like "My mother is not a rational number" is true or false. The formal context of "sets of reall numbers" merely says the relation "is an element of" is a well defined relation between a real number and a set of real numbers. It doesn't say that the relation "is an element of" is defined between something that is not a real number and a set of real numbers.

The notations in the original post appear to indicate a special situation when a context is defined in terms of another context. For example, let context U1 be sets who elements are people. Let context U2 be sets whose elements are sets of people in math class. For example, a set in U2 is "The rolls of all math classes that begin at 8 AM". If some institution actually pays an instructor to conduct a class with no students in it then the null set of sets of people is an element of U2.

(Logician's may have better terminology than "context" to express what I'm describing !)

 

[caveman1917]

(Logician's may have better terminology than "context" to express what I'm describing !)

"Interpretation". Sometimes "model" or "structure" depending on specifics.