Can you define a set by what it excludes?

[G037H3]

S, the set of all paintings. Some subsets would be:

{s|s is a Renaissance piece}

{s|s is monochromatic}

{s|s is painted on oak}

Is it okay to define a subset of a set by what it *isn't*?
Ex:

{s|s was not painted by Picasso}

 

[Office_Shredder]

If S is the ste of paintings, then {S|S is a Renaissance piece} doesn't mean anything. S is never a Renaissance piece because S is the set of all paintings. You need to write this as
{s in S| s is a Renaissance piece}.

Then {s in S| s is not painted by Picasso} makes sense, it's the set of all paintings that were not painted by Picasso. Compare this to {s| s is not painted by Picasso}. Without specifying a set which s has to belong to, this isn't a subset of S anymore. For example, I was not painted by Picasso, so I belong to this set

 

[G037H3]

If S is the ste of paintings, then {S|S is a Renaissance piece} doesn't mean anything. S is never a Renaissance piece because S is the set of all paintings. You need to write this as
{s in S| s is a Renaissance piece}.

Then {s in S| s is not painted by Picasso} makes sense, it's the set of all paintings that were not painted by Picasso. Compare this to {s| s is not painted by Picasso}. Without specifying a set which s has to belong to, this isn't a subset of S anymore. For example, I was not painted by Picasso, so I belong to this set

Okay. As for the notation, I agree. The set notation is adapted from Principles of Mathematics (which I just received and I've started on, during breaks from Python). The text that I adapted to my example states, verbatim (other than the hash comment):

"Subsets of a set A are often defined as containing those elements of A which have some property in common. If S represents the universal set consisting of all the students at your university, we may be interested in the following subsets:

{s|s is a girl} # LOL
{s|s is a football player}
{s|s is a member of the EOE fraternity}
{s|s is a graduate student}

"

I think your issue with how I notated my example is that you take the universal set to be everything, whereas I implicitly assume that it is just of paintings?

 

[Office_Shredder]

The notation of {s in S| s satisfies a property} is often dropped to just read {s| s satisfies a property}, when the larger set is obvious from context. That's not really that big a deal, but I wanted to point out the possibility for confusion. My main concern with your notation (though maybe I wasn't very clear in my post) is the difference between
{s| s is a Renaissance piece} and {S| S is a Renaissance piece}, because S is already defined to be a set, so you shouldn't use it to mean something else.

The reason why it's good to emphasize the larger set when your property is a negative property is because just writing
{s| s is a Renaissance piece} readily restricts s to be a painting (or I guess a statue or something else made in the Renaissance in this case, but usually it's clear) whereas {s| s is not a Renaissance piece} doesn't. It's still clear from the context that you're talking about paintings of course.

The problem can occur when for example I give you
{x| x²=-1}. Without telling you where x can exist, the answer is either the empty set or {i, -i}.

Back to the main question though, of course the property that x must satisfy can be of the form "x is not a ____". In fact, {x in A| x does not satisfy property P} is equal to A-{x in A| x satisfies property P} where the - here is set subtraction

 

[G037H3]

oops...the s's in my examples of subsets are supposed to be lowercase

I apologize.

As for your post:

I was reasonably sure that it is permissible to define sets by what they aren't. My mental imagery of a set is a space, and subsets can only be defined by cutting up a set, obviously. To define one part of the set (space), it is necessary the exclude the rest, so of course it is possible to define a subset of a set by excluding (a) different subset.

 

[G037H3]

I have an additional question about a specific matter, but I don't want to make a new topic just for it:

Using the set R (reals):

{x|x squared is greater than or equal to 0}

It would be {R}, correct? o.o

 

[chiro]

S, the set of all paintings. Some subsets would be:

{s|s is a Renaissance piece}

{s|s is monochromatic}

{s|s is painted on oak}

Is it okay to define a subset of a set by what it *isn't*?
Ex:

{s|s was not painted by Picasso}

When you're defining a set you need to take into account the "Universe" of the set. With regard to your question the set you are looking for is the complement of some set S which contains all the things you don't want.

It makes no sense to talk about a set without defining the universal set because you have nothing to reference against.

 

[Office_Shredder]

Using the set R (reals):

{x|x squared is greater than or equal to 0}

It would be {R}, correct? o.o

{R} is a set containing a single element, which is the set of real numbers. R is a set containing every real number as an element. They're distinct objects.

What you have is R, which is what I assume you intended to say

 

[disregardthat]

It makes no sense to talk about a set without defining the universal set because you have nothing to reference against.

... which is because a set of all sets in set theory is contradictory. Classes of sets however can be defined by only specifying the (defining) property of its elements. Thus all sets are classes, but some classes are not sets (e.g. the class of all sets).

So informally it makes sense to talk about a class of all objects which are not painted by picasso. Classes does naturally not have the set-theoretical properties sets have.

Note that all such classes have a common reference class, namely the class of all sets.

 

[G037H3]

{R} is a set containing a single element, which is the set of real numbers. R is a set containing every real number as an element. They're distinct objects.

What you have is R, which is what I assume you intended to say

Yes, I mean that any real satisfies the condition. But why is {R} different than R? It seems to me that a set is like putting boundaries on elements, so if the boundaries only surround one element, it seems that it *is* the element.

 

[disregardthat]

Yes, I mean that any real satisfies the condition. But why is {R} different than R? It seems to me that a set is like putting boundaries on elements, so if the boundaries only surround one element, it seems that it *is* the element.

{R} denotes the set of the (single) object R, it does not care about its elements.

 

[G037H3]

{R} denotes the set of the (single) object R, it does not care about its elements.

But R represents the reals.

So basically, set theory disallows referencing at a depth greater than 1?
Why?

 

[CRGreathouse]

So basically, set theory disallows referencing at a depth greater than 1?

No no no no no no no!

But it does distinguish them: R has uncountably many elements while {R} has only one.

 

[G037H3]

No no no no no no no!

But it does distinguish them: R has uncountably many elements while {R} has only one.

Then how is what I said wrong? X_X

Okay, {R} only has one element, but the element refers to other elements.

It isn't simple. It's a *composite* element.

 

[Office_Shredder]

composite element doesn't mean anything. The key idea here is that sets are objects also. {R} is a set with only one object in it. R is a set with uncountably many objects in it. They are different.

This is an important distinction. For example, one way the natural numbers are constructed in set theory has that zero is the empty set (which I will call 0), then 1={0}, 2={0,1}, etc. So it's important that {0} and 0 are very different sets

 

[G037H3]

So,

composite elements have no meaning within a set, because the fact that it is compound only pertains to how it affects its relationship with other elements, and not on how the composite element is classified as an element of a set?

the set {R}, it's a collection of one element. A collection of 1*x is not a collection, it is the x itself.

I still don't understand why {R} != R

:/

 

[Office_Shredder]

So,

composite elements have no meaning within a set, because the fact that it is compound only pertains to how it affects its relationship with other elements, and not on how the composite element is classified as an element of a set?

I don't understand what you're saying here. Words like "the fact that it is compound" and "composite element" are not standard terminology, and while I can guess at what you're trying to say it would help if you spell out exactly what you mean

A collection of 1*x is not a collection, it is the x itself.

I don't know what you mean by 1*x here, or how it's a set

I still don't understand why {R} != R

{R} is a set with one element. Is R a set with one element? How can two sets be the same if they have a different number of elements?

 

[CRGreathouse]

Edit: Office_Shredder posted an almost identical response above while I was composing mine.

composite elements

This is your term, not a standard math term. How do *you* define it? We can't address your questions about it without knowing what you want it to mean.

I still don't understand why {R} != R

Sets are equal if and only if they contain the same elements. R does not contain R an an element, so it is not equal to {R}, which does contain R as an element. R contains 0 as an element, so it is not equal to {R} which does not contain 0, etc.

Alternately: Sets with different numbers of elements are never equal. {R} has cardinality 1 while R has cardinality beth_1 which is vastly larger. Thus the two are different.

 

[G037H3]

This is your term, not a standard math term. How do *you* define it? We can't address your questions about it without knowing what you want it to mean.



Sets are equal if and only if they contain the same elements. R does not contain R an an element, so it is not equal to {R}, which does contain R as an element. R contains 0 as an element, so it is not equal to {R} which does not contain 0, etc.

Alternately: Sets with different numbers of elements are never equal. {R} has cardinality 1 while R has cardinality beth_1 which is vastly larger. Thus the two are different.

Composite element = an element made up of simpler elements

R in {R} is a composite element because R represents all of the reals, which are elements in themselves, as R is the set of them

Simple element = an element which is not composite

a specific number, as opposed to a collection of numbers

I halfway understand what you're saying, but it isn't really appealing to my intuition

My intuition says that R = {R} because {R} is meaningless without R, and R is nothing without itself, so, {R} does contain one element, R, that represents all the elements of R

so how is what I said about depth wrong?

 

[Office_Shredder]

Composite element = an element made up of simpler elements

How can I make this judgement? For {A} is A a composite element? Set theory shouldn't depend on what the objects contained inside of your sets are.

In axiomatic set theory generally the only objects are sets, and numbers are defined to be certain sets. Even if you're not going to do that kind of stuff it's a good idea to get familiar with the idea of sets contained inside of other sets as objects just like everything else

My intuition says that R = {R} because {R} is meaningless without R, and R is nothing without itself, so, {R} does contain one element, R, that represents all the elements of R

R represents all the elements of R, but just because 1 is an element of R doesn't mean that it's an element of {R}. Nowhere is it stated that an element of R is contained inside of {R}. When you write a set like {R}, you've explicitly listed its elements, and it has only one. The object that it contains also happens to be a set, but that's just a coincidence and has no bearing on the fact that {R} is a set with a single element. The set {R} has no knowledge of the fact that R has 0,1,2,3,4, etc. as elements of it

 

[CRGreathouse]

Composite element = an element made up of simpler elements

Your definition is circular: composite elements are "made up of" (contain as elements?) simple elements, and simple elements are those other than composite elements.

Perhaps I can interpret "simple elements" as ur-elements, in which case neither R nor {R} are simple or composite in ZF(C).

My intuition says that R = {R} because {R} is meaningless without R, and R is nothing without itself, so, {R} does contain one element, R, that represents all the elements of R

Well there's your problem. "=" denotes equality, not "is meaningless without".

so how is what I said about depth wrong?

You said "set theory disallows referencing at a depth greater than 1", which is clearly wrong. For example, I could say "all elements of {R} are equal to R", which is the same as "For any element S of {R} and for any element s of S, there is an element r in R such that r = s; for any element S of {R} and for any element r of R, there is an element s of S such that s = r." These reference at "depths greater than 1", as you put it.

 

[G037H3]

I should go to sleep...responsibility says so!

I'll try to respond tomorrow, if I'm not too busy. :)

 

[chiro]

I should go to sleep...responsibility says so!

I'll try to respond tomorrow, if I'm not too busy. :)

One way of thinking about it is defining every set so that its like a present or a russian doll.
Lets take two example sets S and T.
S = {0,1,2}
T = {7,8,9}

So we take S and get '0' '1' and '2' and put them in Russian doll 1 and put '7' '8' and '9' in Russian doll 2.

Now we have the set X = {S,T}

For X we put the two previous Russian dolls in another Russian doll.

So to recap for first two sets we have two Russian dolls and for last set we put those Russian dolls inside new doll.

So when we get the elements out of our X Russian Doll, we get out two more Russian Dolls instead of the elements '1', '2', '3', '7', '8' or '9'.

Hope that helps!

 

[G037H3]

One way of thinking about it is defining every set so that its like a present or a russian doll.
Lets take two example sets S and T.
S = {0,1,2}
T = {7,8,9}

So we take S and get '0' '1' and '2' and put them in Russian doll 1 and put '7' '8' and '9' in Russian doll 2.

Now we have the set X = {S,T}

For X we put the two previous Russian dolls in another Russian doll.

So to recap for first two sets we have two Russian dolls and for last set we put those Russian dolls inside new doll.

So when we get the elements out of our X Russian Doll, we get out two more Russian Dolls instead of the elements '1', '2', '3', '7', '8' or '9'.

Hope that helps!

That still seems like the depth 1 issue that I stated before.

As for 'composite elements', I realized that I meant a set within a set, simply within the context of being an element like the other objects inside of a set.

 

[CRGreathouse]

As for 'composite elements', I realized that I meant a set within a set, simply within the context of being an element like the other objects inside of a set.

If you knew the standard set-theoretic construction of the natural numbers you'd know why everyone else here finds that problematic at best. For example, in that construction, 1 ∊ 2 and so it seems that you would call 2 a composite set.

That still seems like the depth 1 issue that I stated before.

I recognize that you have a name for this sort of thing, but I'm not convinced that you understand it.

 

[G037H3]

I've discovered that my prior confusion was based on assuming that class and set refer to the same concept. My statements on 'depth' evoke classes.

 

[disregardthat]

I've discovered that my prior confusion was based on assuming that class and set refer to the same concept. My statements on 'depth' evoke classes.

Every set is a class however, so classes which are sets obviously cannot have properties sets don't have.

 

[G037H3]

Every set is a class however, so classes which are sets obviously cannot have properties sets don't have.

A class is a set of sets, so {3} for instance is a set, not a class. So what do you mean? >_>

 

[disregardthat]

A class is a set of sets, so {3} for instance is a set, not a class. So what do you mean? >_>

3 is a set in set theory, often constructed as such: http://en.wikipedia.org/wiki/Natural_number#A_standard_construction hence {3} is a class. All sets are classes.

Every object formalized in set theory is a set, so the moment you talk about things which are not sets you are stepping outside formal set theory.

 

[AlephZero]

A class is a set of sets, so {3} for instance is a set, not a class. So what do you mean? >_>

The theory of sets which all have a finite number of elements is simple, and doesn't need the idea of "classes".

Classes were defined to sort out the paradoxes that arise with infnite sets. The earliest record of such a paradox (quoted by St Paul in the Bible) is the statement "A Cretan told me that all Cretans are liars." That can't be either true or false, but it looks like a meaningful English sentence, so what's going on here?

At a more abstract level, consider whether "the set of all sets that are not members of themselves" is or is not a set. If it is a set, then by definition it isn't. If it isn't a set, then by definition it is.

One solution to these issues is to carefully define the concept of a set, such that statements like the above are not true or false, but meaningless. The consequence of this is that the idea of "the set of all sets" is meaningless. So you need a new concept called a "class", such that "the class of all sets" is meaningful. See Russell and Whitehead's "Principia Mathematica" for a very great deal more on this. (Warning: it takes them about 1000 pages to get as far as proving from first principles that 1+1=2)

 

[G037H3]

The theory of sets which all have a finite number of elements is simple, and doesn't need the idea of "classes".

Classes were defined to sort out the paradoxes that arise with infnite sets. The earliest record of such a paradox (quoted by St Paul in the Bible) is the statement "A Cretan told me that all Cretans are liars." That can't be either true or false, but it looks like a meaningful English sentence, so what's going on here?

At a more abstract level, consider whether "the set of all sets that are not members of themselves" is or is not a set. If it is a set, then by definition it isn't. If it isn't a set, then by definition it is.

One solution to these issues is to carefully define the concept of a set, such that statements like the above are not true or false, but meaningless. The consequence of this is that the idea of "the set of all sets" is meaningless. So you need a new concept called a "class", such that "the class of all sets" is meaningful. See Russell and Whitehead's "Principia Mathematica" for a very great deal more on this. (Warning: it takes them about 1000 pages to get as far as proving from first principles that 1+1=2)

I'll probably read that book sometime in the next year, after I learn a lot more formal logic.

I already know about the universal set issues (in fact, the book I'm using tries to use the label of a 'universal set' U that contains all possible answers to a problem, when really they mean the 'universe').

The solution to issues such as the above is solved by metalogic, obviously.

I stated an issue with 'depth' from before; this is because I was thinking in terms of CS. For instance:

>>> a_list = ['a', 'b', 'c']
>>> a_list.extend(['d', 'e', 'f'])  
>>> a_list
['a', 'b', 'c', 'd', 'e', 'f']
>>> len(a_list)                     
6
>>> a_list[-1]
'f'
>>> a_list.append(['g', 'h', 'i'])  
>>> a_list
['a', 'b', 'c', 'd', 'e', 'f', ['g', 'h', 'i']]
>>> len(a_list)                     
7

As you can see, when using append, the length is of objects at depth 1, the list ['g', 'h', 'i'] is only viewed as a single object. Before knowing of this, I applied similar thinking to the definition of a set within a set.

 

[Jimmy Snyder]

There is a notation (in fact more than one) in common use for defining a set by what it excludes. A = U \ B, where U is the universal set, and B is the set of things to be excluded defines A by what it excludes. An even simpler example is the empty set which can be defined as the set that excludes everything.

 

[Studiot]

I don't know why G037H3 is struck out since this is my first viewing of this thread.

So here is a simple example about your difficulty with the question

Is {R} $$\equiv$$ R ? (Note I have used the identity symbol, not equals.)

Consider a heap of auto components. An auto contains 1 each of these components in a specific order.

Compare the above question about the reals with the question

Is {one auto} identical to that heap of components?

@Jimmy Snyder

An even simpler example is the empty set which can be defined as the set that excludes everything.

This definition runs into immediate difficulty because the complement of this empty set is the set of everything, which is of course a set of all sets, which some exclude as not a valid set.

 

[disregardthat]

This definition runs into immediate difficulty because the complement of this empty set is the set of everything, which is of course a set of all sets, which some exclude as not a valid set.

The complement with respect to what?