# What is an Empty set?

We define the term "Set" as,

A set is a collection of objects.

And an "Empty set" as,

An empty set is a set which contains nothing.

First problem I encountered:

How the definition of "Empty set" is consistent with the definition of "sets" if "Empty set" contains nothing and a "set" is a collection of objects.

Further we discovered in set theory that every set has a subset that is the Null set.

Such as,

If $A=\emptyset$ and $B=\{1,2,3\}$ then,

$$A \subset B$$

Second problem I encountered:

How and why "No element" is referred and considered as an element as we do in case of null set that is, when we say that every set has a subset that is the Null set?

Third and Last one:

How can a set possess "some thing" and "nothing" simultaneously that is when we say that every set (containing objects) has a subset which is Null set (contains nothing)?

• I hate these silly question huge-amount-of-upvote questions. The first sentence is wrong, we don't define sets as collections of objects, we talk about them like that so people can grasp them intuitively before actually studying set theory. To sidestep defining sets we actually say "suppose an empty set exists" and some other axioms (but none about sets of stuff existing, just the existence of things like union) and go from there. – Alec Teal Apr 28 '15 at 12:26
• You often get into trouble if you try to do math based on dictionary-like definitions like these. Another example: Look up the definition of "number" in the dictionary, then try to prove that 2+2=4 based only that defintion. It would be impossible. – Dan Christensen Apr 28 '15 at 16:49
• What can I do? These definitions are given in the book I am following now a days that is, College Algebra fifth edition by Raymond A Barnett and Micheal R Ziegler If you don't believe me you can go through it. And there you will find the same definition of Set as I stated above. – Sufyan Naeem Apr 28 '15 at 18:23
• @AlecTeal Aren't things so simple when you already have an advanced education in mathematics? Unfortunately for some mere mortals, they have yet to be educated yet and come here seeking to remedy that. A good community welcomes all honest attempts to learn. – dalastboss Apr 28 '15 at 18:29
• @SufyanNaeem No one's saying they "don't believe you"; they're just saying that if you're having trouble understanding that definition (and yes, not understanding why the empty set is a valid set means you're having trouble with the definition), then you should consider some other definitions, or reconsider the definition without trying so hard to make it fit your preconceived notions based on non-mathematical language. – Kyle Strand Apr 28 '15 at 18:40

A shopping bag is an object to carry things; an empty bag is a bag with nothing inside it.

From the viewpoint of people trained in mathematics, the explanation "a set is a collection of objects" is formally consistent with the set being empty (in which case the set is a collection of no objects).

But perhaps, at first, you should just take "a set is a collection of objects" as an informal idea of what sets are. Apart from the empty set, that phrase also has problems with collections of objects that are not sets because they are "too big", e.g. the collection of all sets is not a set.

Moreover, most of the sets considered in set theory aren't really sets of "objects" - they are sets of other sets. In the commonly studied set theories, there are no objects other than sets. This is another way that "a set is a collection of objects" can give the wrong impression.

So don't get hung up on the "a set is a collection of objects" phrase. Once you spend some time working with sets, you will have a better sense of what they are and how they work.

• But since everything inherits from Object, a set must be an object.....oh sorry, I let my Java get in the way :) – Brian J Apr 28 '15 at 13:41
• @SufyanNaeem If the analogy of "a bag is still a bag even when it's not holding anything" doesn't help, I'm not sure what will help. It's unclear what is actually causing the confusion here, since your question simply says that the definition appears to contradict particular examples, but the shopping bag example shows why the definition isn't contradictory. – Kyle Strand Apr 28 '15 at 17:59
• @SufyanNaeem I think my response holds even if you replace "help" with "help much." In any case, in my answer, I did my best to provide comprehensive answers to each of your questions; did that help at all? – Kyle Strand Apr 28 '15 at 20:56
• I think it's clearer to describe the empty set as "a collection of zero objects" than "a collection of no objects". "Zero" makes it clear that you're counting the number of objects in the collection, whereas "a collection of no objects" sounds like a negation of the definition which is, I think, the source of the confusion. – David Richerby Apr 28 '15 at 21:45
• @ruakh: I did not write "a bag of groceries" or a "bag of shopping". I wrote a "shopping bag" and an "empty bag", i.e. an empty shopping bag. A "shopping bag" is just a kind of bag, it is not a "bag of shopping", and I chose my terms carefully. I could remove the word "shopping" entirely from my answer, and the point would remain the same. – Carl Mummert Apr 30 '15 at 1:58

In addition to the answer by Carl, there's a big difference between having the empty set as a member $\emptyset \in A$ and having the empty set as a subset $\emptyset \subset A$. I think you are confusing the two concepts.

From a set $A$ we can create a subset by "picking out" elements of $A$. Now, if we don't pick out anything, we're left with the nothingness we started with, $\emptyset$. It was not a member of $A$, it's another set.

Now, with for example $B = \{ \emptyset, \{a\}, \{b\}, \{a,b\} \}$, we do have that the empty set is a member of $B$. This $B$ could be the set of all subsets of $C = \{a,b\}$, and one of those subsets would be the empty set, as we said before.

• Just as there is a differne between putting a bag (and its content) into another bag or putting the contents of the first bag into the second. (Car Mummert's shopping bag analogy is just to good) – Hagen von Eitzen Apr 28 '15 at 15:02

My usual advice to students that have some hard time in introductory to set theory, is to work with the formal definitions until they develop some intuition.

So I strongly suggest that if you're unclear as to what is the empty set, and what sort of concept it is, you'll sit down and work with its formal definition.

Definition. We say that a set $A$ is empty, if $\forall x(x\notin A)$. Equivalently, $\lnot\exists x(x\in A)$.

Axiom. There exists an empty set, denoted by $\varnothing$.

Assuming the axiom of extensionality, we can show that any two empty sets are equal, hence the empty set.

Claim. If $A$ is a set, then $\varnothing\subseteq A$. In other words, $\forall x(x\in\varnothing\rightarrow x\in A)$. And in English, every element of the empty set is an element of $A$.

Proof. If $\varnothing\nsubseteq A$, then $\lnot\forall x(x\in\varnothing\rightarrow x\in A)$ is true, which means $\exists x\lnot(x\in\varnothing\rightarrow x\in A)$, and equivalently $\exists x\lnot(x\notin\varnothing\lor x\in A)$, which again translates to $\exists x(x\in\varnothing\land x\notin A)$. In particular, $x\in\varnothing$ which is a contradiction since $\lnot\exists x(x\in\varnothing)$ is the definition of the empty set. $\square$

In simpler words, the proof is saying, if this wasn't the case, you should be able to find a counterexample, which means an element of the empty set which is not an element of $A$. But there are no such elements. So the statement holds.

Once you sit to prove a few statements about sets, once you've gone through several vacuous truth sort of arguments like above, once you've meddled with sets, you'll get some picture, and what is an empty set will be clearer.

• I was expecting an answer from you! :) How can a man miss it if he is an upcoming set theorist! – Sufyan Naeem Apr 28 '15 at 19:38
• Well, a man has to teach, a man has a cold he cannot seem to shake off, and a man was not sure what to add to Carl's excellent answer. – Asaf Karagila Apr 28 '15 at 19:40
• TBH, this just defines set from ∈, which is somewhat self-referential if you consider x ∈ Y as x is an element in the set Y. IOW, if you come from the direction of the topic starter, then "set" is a more fundamental concept than "membership of a set". – MSalters Apr 29 '15 at 12:33
• @MSalters: No, this is not "self-referential" at all. The formalist version would be to say, we have a binary relation symbol $\in$, and we have a list of axioms. Given a model of these axioms, we call the objects there "sets". But there is no "formal definition" of the term set which is any different than "an object in a universe of set theory". I agree that coming as a newbie, the objects exist before the relation does. Which is why I suggested to work formally with the definitions and axioms. – Asaf Karagila Apr 29 '15 at 16:35
• Since you've taken the existence of (an) empty set as an axiom above, it might also be worth pointing out that there's nothing fundamentally unprovable about it; i.e. that, in set theories including some form of specification, the existence of an empty set follows straightforwardly from the existence of any set. – Ilmari Karonen Apr 29 '15 at 17:06

Mathematicians don't do grammar; Or more correctly, those who are pedantic about grammar usually can't be precise about mathematics. Unfortunately you have fallen into the trap of the 'common idiom', which is defined by its inconsistencies.

You will note that the Romans didn't have the number (symbol) zero, and much of English is based on the Victorian assumption of a Latin grammar, so you are stuck with what the software coders call an 'off by one' error. No things is a number of things; that number is zero, despite the confusions of common (miss)understanding. A similar trap occurs between 'Or' and 'exclusive Or', with the English version of the former meaning the latter!

It takes a while to become comfortable with these differences and distinctions in mathematical writing.

• Philip, you make a good point - namely, that mathematicians use language differently when they're speaking math, and that this is intentional - but its not phrased to my liking. I'm upvoting, but grudgingly. – goblin Apr 29 '15 at 6:09
• I took a long time working my way through the $\delta-\epsilon$ definition of continuity. It is, in my opinion, one of the finest definitions in Mathematics. But it took me days to see how simple it was. Then I took topology and found out how much simpler continuity could be.I love how simple ideas have such amazing consequences. You need language to communicate mathematical ideas. But the words are a catalyst, not a scaffold. – steven gregory Apr 29 '15 at 20:18
• @Philip I always wondered if 'Syntactic Structures' could be considered a mathematics book. – steven gregory Apr 30 '15 at 0:06

There are some issues with the intuitive definition of a set which you have taken as the basis for your understanding.

It is possible to say "a set is a well-defined collection of objects". What do we mean by "well-defined"? - Well that is the whole question of the foundations of set theory. In studying the question, mathematicians have found it necessary to have some control over the nature of the "objects" in the "collection" - as always with mathematical definitions, we have to be absolutely precise. So mathematics proceeds by building standard sets as a model.

Set theory thus provides a model of standard sets. We can then talk about general sets as being collections which can be put in $1-1$ correspondence with one of the standard sets. We have to get the "objects" of our "collections" from somewhere, as well as language for talking about them - the elements of a group, or the vertices of a graph, or the letters of an alphabet for example. For the set theory model to work, the language we use of our objects has to be compatible with the language we use about sets in the model. Otherwise we need a different model.

Suppose we are solving equations. We may want to talk about the properties of the solutions before we know what the solutions are. Indeed, we may eventually discover that there are no solutions. So we want our model to be sufficiently flexible to deal with this case - and the empty set does the job, we don't have to put a caveat in every sentence. And that's just one example. It is just very convenient to have the empty set as part of the model.

To go back to the equations, we may prove at an early stage that any solution is a positive real number, and use this to derive a contradiction (so there are no solutions). The empty set validates the statement "any solution is a positive real number" and makes it mathematically viable.

• When you say "standard set" do you mean an element of the von Neumann universe $V$ that is built up inductively from $\emptyset$ by repeated application of the power set operation (and taking unions at limit stages)? In most set theories, such as $\mathsf{ZFC}$, everything is a standard set in this sense. (There are alternative set theories with urelements but I'm not sure if this is related to your point.) – Trevor Wilson Apr 30 '15 at 2:41
• In any case, I think it's a bit misleading to say that set theory is used to formalize mathematics by taking a model consisting of standard sets and then finding bijections between collections outside the model and sets inside the model. Where would the bijections themselves live, and how would we reason about them? The usual way to use set theory to formalize mathematics is to adopt the axioms of $\mathsf{ZFC}$ (which means making the tacit assumption that every object is a set) and then to define real numbers, functions, manifolds etc. as objects (sets) satisfying certain properties. – Trevor Wilson Apr 30 '15 at 2:52
• @TrevorWilson But we can talk about sets of shoes and socks and "collections" of physical objects. In this context set theory is definitely a model rather than the reality. I deliberately didn't say we formalise mathematics within set theory, and I said we need "language" to talk about things - and that would include "bijections", though that was not explicit. It was tempting to write more fully about foundational issues, but that did not seem to be consistent with the level of existing knowledge shown by OP. It will be useful to OP to know that the issues run deeper. Thanks for your comments. – Mark Bennet Apr 30 '15 at 6:00
• Ah, I think I was taking "model" and "1 − 1 correspondence" more literally than you intended. – Trevor Wilson Apr 30 '15 at 6:05

### The empty set (first "problem")

The empty set (sets are uniquely defined by their elements, so there aren't multiple "empty sets") is a mathematical primitive--a conceptual entity upon which a larger mathematical framework is built.

Mathematical primitives are notoriously difficult to define. One of Euclid's definitions of a "point" is "that which has no part"; one of my math textbooks snarkily asked "could this not also apply to an out-of-work actor?" Similarly, a "number" can usually be thought of as "how many of something" there is, but in that sense, the number "zero" represents "none of something," which is "weird" in the same way that "a collection of no objects" is "weird."

With that said, there are a few ways of trying to conceptualize the empty set:

• The aforementioned "grocery bag" analogy is pretty good, because it shows that the fact that sets are defined by the fact that they can have mathematical "objects" in them is distinct from the fact that not all sets actually have a non-zero number of objects in them. You can go a little bit further with this: just as you can remove items from a grocery bag until it is empty, you can imagine subtracting a set from itself, i.e., creating a new set with none of the items from the original set--and of course this is still a set, because what else could it be?
• Somewhat similarly, but without the analogy, consider the intersection set operation on two disjoint sets. Sets are disjoint when they contain no common elements, e.g. {A,B} and {C,D}, and the intersection of two sets is the set of common elements. E.g., for {A,B} and {B,C}, the intersection is the set {B}. But for {A,B}, and {C,D}, the intersection is, of course, the set {}.
• Even if the intersection operation doesn't make intuitive sense to you when there are no elements left in the resulting set, you can consider all set operations to be merely text-operations based on grammatical rules. (This is a highly formalized and well-developed branch of math that ultimately underpins much of computer science, but I'll try to be fairly non-technical.) Consider a text-based representation of sets wherein the pair of symbols { and } denote a set, and the symbol , separates each set member from the next. You don't need to "understand" what this "means" in a philosophical sense in order to recognize that, for instance, {A} is a set containing the single element A, and {A}} is an ill-formed (i.e. invalid) textual representation in this scheme ("scheme" here is a non-technical word; the correct word is "grammar"). In other words, {A}} doesn't mean anything; it doesn't represent a set, because } must always be paired with {. (Note that I was implicitly using this grammar in the previous bullet point without needing to explain it; it is, of course, fairly intuitive.) Now, is {} a valid set in this grammar? The answer is yes, because the braces are correctly paired. What elements are contained by {}? There aren't any.
• The formal set-theoretic way of defining sets is actually to start with the empty set, define set operations, and permit including sets within other sets (i.e. state that for every set S, {S} is also a set). Here, there is no possibility of the existence of {} contradicting the definition of the word "set", since our definition of "set" starts with the words "{} is a set."

### Subsets (second "problem")

You seem to be getting the wrong impression from the word "subset". The sentence "A is a subset of B" does not imply either of the following:

• A is "smaller than" B (i.e. has fewer members)
• Sometimes the phrase "proper subset" is used to indicate a set that is "smaller than" the superset: i.e., if A is a proper subset of B, then there exists at least one element of B that is not in A.
• A is a member of B (i.e. A is one of the "objects" in B)

All that sentence means is that, for every "object" that is in A, it is also in B. There are two easy ("trivial") ways for this to be true:

• A is B. That is, they contain exactly the same elements. If A is B, then is it possible for A to contain something that B does not? No, of course not. Thus A is a subset of B.
• A has no elements. If A has no elements, then does it have any elements that are not in B? No, because it does not have any elements, period.

The second bullet point, obviously, is why the empty set is a subset of all sets. The first bullet point is the special case describing the fact that {} is a subset of itself. Here, where A is {} and B is also {}, it is both true that A has no elements (and therefore has no elements that are not also in B) and A and B contain exactly the same elements (i.e., neither contains any elements). Thus, trivially, "all elements" in A (all none of them!) are also in B.

If it confuses you that statements are considered "true" when they don't actually describe anything, consider the following statement: "All unicorns lack horns." Instead of asking why this is or isn't true, consider how it could possibly be false. If it were false, then at least one unicorn would have a horn. But no unicorn actually exists, and so there are no unicorns that actually have horns! Thus the statement cannot be false. Similarly, it is also true that "all unicorns have horns", because we are simply making blanket claims about nothing. There are no unicorns that lack horns, so all unicorns have horns.

### Containing something and nothing (third "problem")

This is really just more discussion about the concept of subsets. Once again, is a subset of does not mean is a member of.

Consider a set S described as {A,B,C} (i.e. its members are A, B, and C, whatever those are).

It "possesses something," in your words, because it "possesses" (i.e. "has members") A, B, and C. Now, does it make sense to say that it "possesses nothing"? Well, maybe, but that's incredibly vague terminology, so let's go back to what's actually being claimed: "the null set is a subset of all sets, and therefore the null set is a subset of S."

Does this mean that the null set is a member of the set S? No, because S's members are A, B, and C, and as far as we know, none of those are the null set.

But it does mean that S has within it every member of the null set. (This is just the "inverted" way of saying that all members of the null set are also members of S.) In other words, there are no members of the null set that are not members of S. How do we know this? Because there are no members of the null set... period.

Now, what if S did contain the null set? What would that look like? Well, then we would have S described by {A, B, C, {}}. (Ignore spaces; they don't mean anything.) Now, {} is still a subset of S, because, again, {} has no members that aren't also in S. But now, {} itself is also a member of S. This might seem a bit weird at first, but remember that sets aren't collections of "objects" in the grocery-bag sense of "containing" physical objects; they're collections of mathematical entities or concepts. (Even integers are "mathematical concepts" rather than physical realities; what is the physical reality of, say, the number 3?) {}, as established, is a mathematical entity; it has a definition and a meaning. So the set {A, B, C, {}} isn't any "weirder" than the set {A, B, C, 0} or the set {A, B, C, (0,0)} where (0,0) represents the origin-point of a Cartesian coordinate system.

People used to argue that $0$ wasn't a number for reasons very much like your arguements about the empty set.

Forget about defining sets. The most important thing about sets is this.

If x is an object and S is a set, then either $x\in S$ is true or $x \notin S$ is true; but never both.

When people say that no element is a member of the empty set they are just, badly, trying to say that $x \in \varnothing$ is always false and $x \notin \varnothing$ is always true.

There is a very basic connection between sets and logic. Learning set theory will help you use and understand logic and vice versa.

It's also very convenient to have an empty set. For example.

• If we want to say that sets A and B have no elements in common, then we can say that their intersection is the empty set.
• If we want to say that an equation has no solution, then we can say that the solution set is empty.

There is no nice way to put this. You are confused because you refuse to change your ideas about the way mathematics works. I am very good at mathematics and I have had that same problem more than once. If you are lucky, someone will explain it so that things in your head click into place. If not, which is the more probable thing, you are going to have to work this out yourself.

Set is actually a 'well-defined' collection of objects. Here the adjective well-defined means, such a definition using which, for every element in contention for being element of a set, we can definitively say whether that element belongs to the set or not.

You could ask the same question for 0 - how come zero is a number? The answer is it wasn't for a long time until Brahmagupta decided to make it one. How did he do that? He defined operations of addition, subtraction, and multiplication on zero as they were defined for other numbers. He also extended usual attributes of associativity, distributivity, commutativity etc. And now zero is essential, especially since it serves as the identity element for addition on whole numbers.

Similarly null set is the identity element for the union operation among sets. So, null set is a set as much as zero is a number.

• Never mind zero: for the longest time western mathematicians disputed whether or not one is a number! Euclid defined a "number" as a multitude of units, so 2 would be the smallest number, and it was not until the late 1500s (almost two thousand years later) that consensus finally seemed to come down on the side of 1 being a number. – Henning Makholm Apr 28 '15 at 15:08
• Also, "well-defined collection of objects" doesn't work as a definition of what is or isn't a set. That still leads to nonsensical results, such as the Buralli-Forti paradox (the collection of all possible order types of well-orders cannot be allowed to be a set). – Henning Makholm Apr 28 '15 at 15:11
• @HenningMakholm: And then it repeated again with $\sqrt{-1}$. For quite a while it existed only in the imagination. Even today, some people think that all mathematics exists only in the mind... – user21820 Apr 29 '15 at 5:51
• Look at some of the adjectives used to describe numbers. Negative, Odd, Imaginary, Complex, Irrational... I like to think that a number that isn't a surd, is an absurd number. – steven gregory Apr 29 '15 at 20:22

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