Thread: Set, Subset, and Empty Set

Definitions:
A set is a container and its members (contents).
A sub-set is the container and some, or none, of its members.

Observations:
The empty set is the container. The container is a sub-set; a container is not a member of the set- an egg basket is not an egg.

It is the notion of container that distinquishes set from a collection or group of things. No eggs on the ground is an empty collection of eggs, which is trivial or meaningless. An empty egg basket has meaning.

If the container, such as an egg-basket, of a set is abstracted to “container,” the empty set is a subset of every set.*

Mathematical Definition of subset: U is a subset of S if every member of U is a menber of S, or equivalently, there are no members of U which are not members of S.

Does the mathematical definition hold up for the empty set? Yes: There are no members of the empty set which are not members of S. The empty set is a subset of S.

*EDIT: "the empty set is a subset of every set." is absoluteley wrong. I put correction in next post for emphasis, but thought I'd better correct it here too. The empty set (container) of integers is not the same as the empty set (egg basket) of eggs. You can't abstract the definition.

There is an incorrect statement in previous post serious enough to warrant a separate post:

"If the container, such as an egg-basket, of a set is abstracted to “container,” the empty set is a subset of every set."

Different sets have different containers: An empty set of integers is not a sub-set of a set of letters. Therefore, ” the empty set is a subset of every set." is false.

i appreciate where you're coming from, but sets are not "bags" or "containers" (if only they were...life would be so much simpler!).

a basic rule of sets (parahprased, in lay terms) is: "a set is defined by the elements it contains".

so if a set S contains the elements a,b and c, and NOTHING ELSE, S = {a,b,c}.

the empty subset of eggs, and the empty subset of integers have the same elements (none). therefore, the same set.

perhaps one might say that they are "set-isomorphic" instead (this is a legitimate view, but requires a logical framework LARGER than sets alone).

for the only difference between the "empty bag of eggs" and the "empty bag of integers" is just the LABEL on the bag.

so is { } where the blank space COULD be eggs (but isn't) any different than { }, where the blank space COULD be integers (but still isn't)?

there is a difference between a "thing" and its "description", i can refer to you as Hartlw, or as "the OP", the fact that the descriptions differ doesn't make you "two different people".

an "empty bag" is a description (actually, an analogy) of the empty set, it's not the formal definition.

sets, by definition, are formed by allowing a predicate P to range over some other set, so we have:

S = {x in T: P(x)}

given the existence of ANY set whatsoever, say S, we have:

Ø = {x in S: P(x)} where P(x) is the predicate: P(x) = (x is not in S).

since x can't possibly be both IN S and NOT in S, there must be no such x's at all.

it "doesn't matter" that the x's not in S yet in S, are different kinds of x's than the x's not in T yet in T, or U, or whatever, the point is, we don't have anything in any of these sets.

it seems pointless to belabor this with notation like Ø_S, or Ø_T, etc...all these sets all "behave" the same way (they pretty much sit around and do nothing all day. lazy, i tell you!)

Hello, Hartlw!

You are trying to place labels on empty sets.

represents . . . a set with no elements.

It remains the same regardless of the objects in the discussion,
. . whether they be armadillos or windmills.


By your reasoning, we can never claim that
. . (After all, "no strawberries" is not the same as "no guitars".)

Thanks for your reply Soroban. You have caused me to refine my thinking on sets and empty sets:

Giving a name to multiple objects makes the concept of no members meaningful. The name is the container. “Set,” “Group,” “Collection,” “A” are containers of Set, Group, Collection, A. Brackets are another way of designatiing a set- the brackets are the container.

The empty set is determined by its members. An empty flock of birds can’t be populated with elephants.

Given two abstract sets, an empty set of one is not a subset of the other unless it is assumed or specified that the members are the same.

there is a famous saying: "you can't step in the same river twice."

what do we mean when we say: an orange is a fruit? certainly we don't mean that the terms orange and fruit can be used interchangeably. what we mean is more something like: an orange is an example (instantiation) of a fruit. we understand there may well be other kinds of fruits, such as apples. we implicitly understand the context is "fruit" and the identifier ("which fruit") is "orange".

sets aren't really "determined by their elements". here's a (of necessity, informal) proof:

suppose we have the set {a,b,c}. suppose that to Alice, a = Ted, b = Bob, c = Edward, but to Betty, a = apple, b = banana, and c = coconut. perhaps Alice and Betty talk all day long about the set {a,b,c} and feel they are in complete agreement that they are talking about "the same set", but neither one actually divulges to the other what they are taking a,b and c to actually BE.

there is nothing in set theory that distinguishes {Ted,Bob,Edward} from {apple,banana,coconut}. to be able to tell them apart, we need to know about some "bigger set" like:

{Alice,Betty,Bob,Edward,Roger,Ted,apple, orange,coconut,mango}.

that is: the actual identification of the individual OBJECTS that comprise a set depends on CONTEXT. we indicate this like so:

S = {x in T: property P is true of x}

T is the context (the background set that allows us to actually specify WHICH x we have in S).

now what you are saying, in this light, is something along the lines of: the integers which don't exist (such as odd numbers divisible by 2), are not the same "non-existent things" as the elephants that don't exist (for example, the elephants that are also birds).

here is where the "linguistics" breaks down. let's say i have an empty bag of integers, and an empty bag of elephants. if i remove the "labels" (the context) how can you tell them apart?

sets are an IDEA: the concept of a set is an abstraction, formed from the notion of a collection (which is a very real and concrete thing). in ordinary language, we give names to things to IDENTIFY them. i am Deveno. by this, i certainly do NOT mean i am a string of 6 alphabetical characters. i am giving a symbolic way to "abbreviate": "that guy who is writing this post in mathhelpforum about blah blah blah, and has brown hair, and brown eyes, and so on....".

in mathematics, however, the NAME is NOT the object. it's just a NAME. the object is whatever thing (and there may be MANY such things) satisfying the defining characteristics (properties) we wish the object to have. yes, the cyclic group of order 4 {1,i,-1,-i} is not the same as the set of congruences modulo 4 {[0],[1].[2],[3]}, but "we don't care", it's really only the NAMES that are different.

set theory is a kind of "logical algebra". there isn't any real hope of ever defining once and for all exactly which things are sets, and which things aren't. it's just an IDEA, a way to organize how we THINK about things.

when we say "the empty set", we mean "the only subset (of some other set) with nothing in it". all empty sets are "the same" (an integer which is odd and even is also not: an elephant which is a bird as well), it serves no CONSTRUCTIVE purpose to distinguish between "impossible integers" and "impossible elephants". no possible logical contradiction will arise if we call these empty bags "all the same bag". integers are not elephants, but those integers that are NOT integers may as well be the elephants that are NOT elephants.

it WOULD be bad if an empty set HAD something in it. for such a thing would be a self-contradiction, and as such, would cause god to divide by 0, and destroy the known universe. thankfully, this never happens.

perhaps (at some level) it would be more correct to say: "all empty sets are set-isomorphic". it's important to understand that this means: "set theory" cannot tell them apart. in a wider context (such as discussing mathematical ideas using the whole of human culture as context), WE may be able to tell them apart (by looking at the label on the bags). but that is a philosophical statement OUTSIDE of set theory. in all fairness, no two things with different descriptions are the same:

"1+1" is a string of three symbols with two "1" 's and a "+" in the middle, how can this POSSIBLY be "the same" as "2"? that is a description containing only ONE symbol.

this is typical of abstraction, we are constantly substituting equivalences for equality. why? "because you can't step in the same river twice", if we considered every distinct thing, at every distinct point in time as TOTALLY different (and in truth, they ARE), communication itself would be IMPOSSIBLE.

i urge you to read the lovely paper by Paul Benacerraf, "What Numbers Cannot Be" (The Philosophical Review 74, Jan. 1965). there is a real danger of losing power of expression by being "too concrete".

Let S1 = {Alice,Betty,Bob, apple, orange, coconut }. This is the set (container + members) by definition. Any other set is equal to this set iff it has the same members and the same container. The container requirement means this can’t be the subset of a different set. There are subsets of this set by the ususal definition. But the “container” of both sets is the same. If I add bananna to a subset of S1 I define a new set (container).

Let S2 = {Alice,Betty,Bob, apple, orange, coconut, mango}. Then S1 is either a subset of S2 with the container of S2, or a full Set with its own container, ie, S1 is either a set with container defined by its members, or a subset with a container defined by the mother set.

here's the trouble:

when you say: "if i add banana to S1...."

sets aren't "collections of objects". that is a useful analogy for understanding sets, but that's not what sets ARE. sets are things that follow the axioms of set theory. these behave "like" containers, but you can only stretch an analogy so far.

the formal statement of the axiom of restricted comprehension (the axiom of sets that allows us to specify a set by: {x in T: P(x) is true}) is:



in ordinary language, this says: for any (given) set A, there is a subset B with x in B if and only if x is in A, and φ(x) is true.

in other words, to define B, we need A first.

which begs the question, how do we get "bigger sets" to choose "smaller sets" from?

well, that is another axiom, the axiom of power set:



which says in ordinary language, for every set A, there is a set P (P is the power set we're after) with the following property:

the elements of P are those sets B for which C is an element of B implies C is an element of A.

(this is a fancy way of saying that B is a subset of A).

now, given a set to start with, like say the empty set { }, we can form its "set of subsets". it has only one subset, itself. so the power set of the empty set is:

{ { } } (a bag with another empty bag inside it). let's call this new "double-bagger set" A (just to make things look nicer).

we can now form the power set of A:

{{ }, A} <----this has two elements.

we can keep going, we'll call our new "two-element set" B. here's what the subsets of B look like:

{{ }, A, {A}, B} <---now we have four elements.

each time we do this, we get a "bigger set", so now we can start creating subsets of these power sets, to get sets of various sizes.

there are other ways to make bigger sets, as well, we can form PAIRS of elements, and we can form unions of sets. each method has its corresponding axiom telling us that whatever we get from pairing or unioning existing sets, is thereby ALSO a set.

nowhere in here have i made ANY mention of integers, or elephants, or bananas, or Alice or Bob. in fact, there is nothing in what i have done so far to even suggest that the collection of all elephants in existence is actually a set.

why? because there is a difference between a theory and a MODEL for that theory. a model is like an example you can point to, and say: that there...it's a such and such. a theory is just a bunch of made-up rules. since we're making the rules up, they might not even be particularly GOOD rules.

you seem awfully preoccupied with pronouncing that "this is wrong", or: "this makes no sense". you might be right, but you don't really KNOW why you might be right. let me make an even sharper point:

logic has nothing to do with TRUTH (the english sense of the word). logic is a "rule-system". why this particular rule-system? because it models the way we think we think. we could be wrong, neuroscience is really in its infancy. the rules of logic derived from "classical reasoning" have proved useful as a way to check the consistency of what we think. they allow for a general consensus among mathematicians that facilitates greater precision in communication than natural language. but...the whole ball of wax MIGHT be "ill-founded"....for example, it might be that "sets" are the wrong concept (a more currently in-vogue alternative is "topoi", which can be very strange, indeed).

i humbly submit you are not looking at sets in a very sophisticated manner, but rather naively, and wanting them to obey "common-sense rules". good luck with that.

I have defined in clear and precise terms what a set is, what a sub-set is, what an empty set is, how to enlarge a set, and what equality of sets means. If someone wishes to show that my definitions satisfy or do not satisfy a particular set of set axioms based on behaviour, that’s fine. What do the set axioms say about sets that I didn’t say?

Obviously my illustrrations do not mean that sets consist of birds and elephants. The members can be quite genral as long as the basic rules I set forth are observed.

you have given a definition. but let's look at that a little more closely:

a set consists of a container and its contents. hmm....what is a container? what can be contents? can sets be contents of other sets?

can a set be a container? what, exactly, is the container that contains the set of all elephants (if indeed all elephants form a set)?

your definition is not one i've seen before (did you invent it? did you adapt it from someone else? i'd like to know....).

in current Zermelo-Fraenkel set theory, "sets" are never defined...only properties which sets may possess. there's a good reason for this: the sum total (union) of all sets isn't one. one cannot, in general, talk of the set of all groups, or the set of all fields, or the set of all vector spaces....all of these collections are "too big" to be a set. in fact, no one seems to know just "how big is too big". what IS known, is that certain things are accepted to be sets (usually including the real numbers, for example....but there ARE those who think the real numbers are "ill-defined". that's an entirely different discussion).

don't get me wrong, i'm certainly no expert on set theory. in fact, the entire subject bores me to tears. i find it a "language of convenience" and am prepared to accept its peculiarities insofar as it helps me to say what i really want to.

i'm not sure you understand fully the distinction between A ⊆ B, and A ∈ B. both of these might be expressed by the english phrase "A is contained in B", so i must ask, to which of these predicates does your term "contained" refer?

do you understand the distinction between "set-isomorphic" and "equal"? let me ask you, which of the following sets represent the natural numbers:

N = {{Ø},{{Ø}},{{{Ø}}}},{{{{Ø}}}},etc.} that is, n+1 = {n}

N = {{Ø},{Ø,{Ø}},{{Ø,{Ø}},{{Ø,{Ø}}}, etc.} that is, n+1 = nU{n}

these are clearly "different" sets.....according to set theory, the set N should be unique....(this is a bigger problem than it might appear, at first).

your definition of equality of sets implies (if i understand it correctly, which i may not) that the set of real numbers, {1,2,3}, is different than the set of integers {1,2,3} because they come from "different containers". this does seem to pose a rather serious ontological dilemma....which set should we do mathematics in, so we can rest assured we are talking about "the same things"?

you have given a definition. but let's look at that a little more closely:

a set consists of a container and its contents. hmm....what is a container? what can be contents? can sets be contents of other sets? Container is set definition. Of course sets can be members of other sets, a set can be defined for any objects.

can a set be a container? no, a set is container and members. what, exactly, is the container that contains the set of all elephants (if indeed all elephants form a set)? The container is the definition of the set.

your definition is not one i've seen before (did you invent it? did you adapt it from someone else? i'd like to know....). I invented it because nothing else made sense.

in current Zermelo-Fraenkel set theory, "sets" are never defined...only properties which sets may possess. there's a good reason for this: the sum total (union) of all sets isn't one. one cannot, in general, talk of the set of all groups, or the set of all fields, or the set of all vector spaces....all of these collections are "too big" to be a set. in fact, no one seems to know just "how big is too big". what IS known, is that certain things are accepted to be sets (usually including the real numbers, for example....but there ARE those who think the real numbers are "ill-defined". that's an entirely different discussion). properties are the container.

don't get me wrong, i'm certainly no expert on set theory. in fact, the entire subject bores me to tears. i find it a "language of convenience" and am prepared to accept its peculiarities insofar as it helps me to say what i really want to.

i'm not sure you understand fully the distinction between A ⊆ B, and A ∈ B. both of these might be expressed by the english phrase "A is contained in B", so i must ask, to which of these predicates does your term "contained" refer? The first if A is a subset of B (same container), If B is a set of containers (empty sets with properties), and A is one of the containers, then A belongs to B..

do you understand the distinction between "set-isomorphic" and "equal"? let me ask you, which of the following sets represent the natural numbers: No, I'd have to look up set-isomorphic. Equal means same members and same container.

N = {{Ø},{{Ø}},{{{Ø}}}},{{{{Ø}}}},etc.} that is, n+1 = {n}

N = {{Ø},{Ø,{Ø}},{{Ø,{Ø}},{{Ø,{Ø}}}, etc.} that is, n+1 = nU{n}

these are clearly "different" sets.....according to set theory, the set N should be unique....(this is a bigger problem than it might appear, at first). Don't understand the definition, but if definition (container) and members are the same, they are the same.

your definition of equality of sets implies (if i understand it correctly, which i may not) that the set of real numbers, {1,2,3}, is different than the set of integers {1,2,3} because they come from "different containers". correct this does seem to pose a rather serious ontological dilemma....which set should we do mathematics in, so we can rest assured we are talking about "the same things"?] Not serious at all, choose the set (container) to suit the use. Sets can have same members but they are only equal if they also have the same containers.

Sorry to kind of rush on this. Have to get ready for a Thanksgiving dinner. Will study it again for typos or things I missed.

EDIT: Have to give up studying this. Time to go

The empty set is the container of the set.
It can be designated as follows:

A={ x:description} read set of x such that...

A₀={ _:x:description} read empty set of x such that...

A_S={ description:x:description} read sub-set of x such that...

A = B iff A₀ = B₀

EDIT: REFERENCE: "A set is somethig which has members or is the empty set." Suppes, Axiomatic Set Theory, Dover