Validity of empty sets

As i have read that a set is a collection of well defined objects or elements but empty set means that there is no elements in the set.We say for example "it is a set of cups,a set of pens" and like wise then what is empty set and two empty sets can be different?

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First of all, how do we even know that an empty set exists? Formally, if we've got some set $C$, then the set defined as $$\{x\in C:x\notin C\}$$ is necessarily empty. (Why?) Let's try to deal with this in a more visualizable way, though.

Imagine a universe in which nothing exists, except for a collection of cups. Since this universe exists in our imaginations, then let's allow ourselves to impose a "force bubble" in this universe to gather up any subcollection of cups that we like. Let's start by putting a force bubble up that gathers all the cups into a bunch. We can think of this force bubble, in a sense, as a set, and the cups as its elements. Now, we'd like to form a different set, one that contains only the things that aren't currently contained in the force bubble we have up right now. To get this set, we simply drop the force bubble already in place, and put up another one somewhere else so that no cups are inside the new force bubble. Since cups are the only thing in this universe, then there's nothing in the force bubble at all--that is, the new force bubble is an empty set. On the other hand, the only things that our force bubble can contain is cups, since there isn't anything else in this universe, so the new force bubble is a set of cups. In this way, we can imagine an "empty set of cups". (We can similarly imagine an "empty set of pens".)

To draw a parallel with the formal version, we started with the set $C$ of all cups (in our universe), then we wanted the set of all things that weren't in the set $C$--that is $\{x:x\notin C\}$--but everything in the universe was a cup (was in $C$), and so the new (empty) set was $\{x\in C:x\notin C\}$.

Consider an "empty set of cups" and an "empty set of pens", to stick with your example element types. Are they the same set? Well, neither set has anything in it, correct? Consequently, the sets are the same, as their elements are precisely the same (vacuously). This is true even though we didn't describe them the same way.

It isn't a problem to describe a set in different fashions, so long as the descriptions give us the same elements. For example, let's say that $A$ is the set of all even integers, and that $B$ is the set of all integers whose squares are integer multiples of $4$. More formally, $$A:=\{n\in\Bbb Z:n=2k\text{ for some }k\in\Bbb Z\}$$ and $$B:=\{n\in\Bbb Z: n^2=4m\text{ for some }m\in\Bbb Z\}.$$ One can show that every element of $A$ is an element of $B$, and vice versa, so despite the different descriptions, they are the same set.

As it turns out, an empty set is a subset of every set. This may give rise to confusion, as one may ask (for example), what a set of cups and a set of pens have in common. The answer is: such sets have no elements in common at all, which is why their intersection (set of their common elements) is necessarily empty. The fact that an empty set is a subset of every set allows us to conclude that there is exactly one empty set--which I'll denote by $\emptyset$.

There are in fact infinitely many ways to describe $\emptyset$. For example, consider the following different ways: \begin{align}\emptyset &=\{n\in\Bbb Z:n=n+1\}\\\emptyset &=\{n\in\Bbb Z:n=n+2\}\\\emptyset &=\{n\in\Bbb Z:n=n+3\}\\\emptyset &=\{n\in\Bbb Z:n=n+4\}\\ &\:\vdots\end{align}

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'the "empty set of cups" and the "empty set of pens" ... Neither set has anything in it, yes?' I suspect the OP was asking whether such sets exist at all ... – Peter Smith Nov 27 '12 at 19:54
Perhaps so. I'll edit accordingly. – Cameron Buie Nov 27 '12 at 20:11
@PeterSmith: Thoughts on my edited answer? – Cameron Buie Nov 27 '12 at 20:58
Well, it just begs the question at issue to suppose that '$\{x\in C:x\notin C\}$' denotes a set: Cantor would have said there is no such set. As to the 'force bubble' metaphor, this doesn't work to illuminate the idea of sets-qua-collections: a collection of pens is one thing, a collection of pens with something wrapped round it is another thing! – Peter Smith Nov 27 '12 at 22:33
The idea I was trying to convey with the force bubble was simply a method of delineation between which objects were in the collection and which objects were not--making it a force bubble rather than an actual substantial bubble (or a box, as I'd initially intended) was intended to avoid the "something wrapped round it" issue. Ah, well. I'm not coming up with any better visualizations at the moment. – Cameron Buie Nov 27 '12 at 23:48

I take the worry behind the question to be as follows: "A set is a collection of well defined objects. OK: but if I have no pens, it would be a mere joke to say that I still have a collection of pens, an empty collection. So how can there even be such a thing as an empty set?" The answers so far do rather beg the question by just assuming that there is such a thing as the empty set.

Now the question why postulate such a thing really does need an answer. The worry has been raised often enough before. For example, it has been said that

• "A class ... consists of objects; it is an aggregate, a collective unity, of them; if so, it must vanish when these objects vanish. If we burn down all the trees of a wood, we thereby burn down the wood. Thus there can be no empty class."

And another author wrote that

• Two point-sets with no point in common lack an intersection (‘sie seien ohne Zusammenhang’). [NB not they have an intersection, the empty set.]

Now don't dismiss these as claims by the thumpingly ignorant! The first is from Frege, the second from Cantor himself (yes, the founder of set theory didn't believe in the empty set). Even Zermelo said that the empty set is ‘not properly speaking (uneigentlich) a set’.

So: there is very good precedent indeed for being worried by the idea of there being an empty set (as the OP is worried).

Of course, we've all got very used to the idea of 'pure' set theory WITH an empty set (as a starter for set-building) and WITHOUT urelements. Modern textbooks boldly assert that that this is the way to go, and wave away beginners' worriers. But arguably doing things this way is to move rather away from the original motivating idea of sets as collections. And it is fact possible -- and arguably rather more natural and more true to the collection idea -- to do things the other way around, with urelements but no empty set.

But of course pursuing that further is too a long story for here: I'm just making the point that it isn't obvious that a useful set-theory has to allow an empty set.

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Hi Peter, I just saw this post. Do you have some references? I would love to mention in class some of these quotes and comments. – Andrés E. Caicedo Feb 28 '14 at 16:08
There are more quotes, if I recall, in Alex Oliver and Timothy Smiley's paper at onlinelibrary.wiley.com/doi/10.1111/j.1520-8583.2006.00105.x/… – Peter Smith Feb 28 '14 at 16:23
And even more in Ch. 14 (on set theory) in their book Plural Logic – Peter Smith Feb 28 '14 at 16:25

There is only one null set (better called the empty set) and it is alternatively denoted with "$\;\emptyset\;$" or "$\;\varnothing\;$" or simply "$\{\,\}$".

The empty set is the set with no elements.

The empty set is unique (there is one and only one empty set) because, by the definition of set equality, two sets are equal precisely if they have the same elements. So If $\varnothing_1$ and $\varnothing_2$ are both empty sets, then they each have NO elements, and hence must be equal, since they have precisely the same elements (namely, none). That is, $\varnothing_1 = \varnothing_2 = \varnothing$.

For added insight, you might want to read the following posts: What is a null set? and Assumption about elements of the empty set.

By the definition of a subset, the empty set is a subset of every set. $\varnothing \subseteq A$ for any set $A$ simply means that IF there exists an $x\in \varnothing$, THEN $x\in A$. This is vacuously true, because it will never happen that there is an $x\in \varnothing.$

To read the definition of "urelement", and a brief introduction to its relation to set theory (expanding on Peter's contribution), see the entry Urelement in Wikipedia. See also the Wikipedia entry on "Empty Set", and "Empty Set: Questioned Existence", in particular.

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But the OP's first question was surely why believe in such a think as an empty set at all? – Peter Smith Nov 27 '12 at 19:53
@PeterSmith You keep me on my toes! (That's a good thing.) – amWhy Nov 27 '12 at 20:03
@PeterSmith - I didn't want to cover ground that you covered so well, so I let most of my answer stand. But I did include some links that expand a bit on your answer, and help to validate the concern expressed in the question. – amWhy Nov 27 '12 at 23:13

By "null set" you seem to mean empty set, and I advise you to use the term empty set instead because "null set" is usually used to mean a set of measure $0$ in some measure space.

Anyway,

The empty set is a set which has no elements. And there is only one empty set.

In general we say that two sets are the same if they have the same elements (this is called extentionality). So if you have two sets with no elements, they are in fact the same set. Thus there can only be one empty set.

A "natural" time the empty set can arise would be when intersecting two sets.

Suppose I have a set $A$ consisting of all pens in my room, and a set $B$ consisting of all dogs in my house. It is a natural idea to consider the intersection of two sets, and in this case that would be

$A\cap B$ the set of all pens in my room that are also dogs in my house.

Of course there can be no elements in $A\cap B$ so $A \cap B$ is the emptyset.

EDIT: As you point out in the comments you can also get the empty set in different ways.

For example, as you say if we had let $A$ be the set of all people and $B$ the set of all pens, certainly $A\cap B$ would have no elements, so it would be the empty set.

The thing that is common to both examples is that the resulting set has no elements. As I said above about when we call two sets the same this means that the sets are the same and are both the empty set. It doesn't matter how we arrived at a set with no elements, as long as it is a set with no elements it is the empty set.

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Then it should be subset of every set but suppose one set is of pens and other is of men.Then it should be intersection of these two sets.I am not understanding what is common between pens and men.Please elaborate more. – Abhinav Anand Nov 27 '12 at 18:31
It was overwhelming to here like this.Lot lot of thanks. – Abhinav Anand Nov 27 '12 at 18:41
@AbhinavAnand glad to help, if anything is still unclear feel free to let me know. – Deven Ware Nov 27 '12 at 18:45
Sure I will do if I will be in serious need. – Abhinav Anand Nov 27 '12 at 18:48