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)?

Solution 1:

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.

Solution 2:

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.

Solution 3:

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.

Solution 4:

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.