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We define the term “Set” as,

A set is a collection of objects.

And an “Empty set” as,

- If $(W,<)$ is a well-ordered set and $f : W \rightarrow W$ is an increasing function, then $f(x) ≥ x$ for each $x \in W$
- Definition of functions
- Why is “for all $x\in\varnothing$, $P(x)$” true, but “there exists $x\in\varnothing$ such that $P(x)$” false?
- Surjective, Injective, Bijective Functions from $\mathbb Z$ to itself
- Partitioning an infinite set
- Recommendation on a rigorous and deep introductory logic textbook

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)?

- The “Empty Tuple” or “0-Tuple”: Its Definition and Properties
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- Proofs of consistency for two formal systems
- Proof that $2^{\mathbb N}$ and $\mathbb R$ have same cardinality
- Do De Morgan's laws hold in propositional intuitionistic logic?
- Cardinality of $\mathbb{R}$ and $\mathbb{R}^2$
- Terry Tao, Russells Paradox, definition of a set
- Proofs involving $ (A\setminus B) \cup (A \cap B)$
- Does there exist a non-empty set that is a subset of its power set?
- Proof that $\bigcap_{a\in A} \bigg(\, \bigcup_{b\in B} F_{a,b} \, \bigg) = \bigcup_{f\in ^AB} \bigg(\, \bigcap_{a \in A} F_{a,f(a)}\,\bigg) $

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.

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.

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.

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.

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.

*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.”

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.

- Sometimes the phrase “proper subset” is used to indicate a set that is “smaller than” the superset: i.e., if A is 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.

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.

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