Why is $\{1,2,3\} \neq \{1,2,3,\emptyset \}?$

Question

I understand why this is the case comparatively , if a set is looked at as being a 'box', then the empty set is considered to be an empty box. So the set $\{1,2,3\}=A$ is considered to be a 'box' containing elements $1,2,3$, and $\{1,2,3,\emptyset \}=B$ is the set containing the elements $1,2,3$ and an 'empty box'?

What I struggle with is how this intuition would make sense given the empty set is a subset of every other set.

Also, I know that $\{\emptyset \} \neq \emptyset$, so would it be the case that for the given set $A$, $\emptyset \subset A$, but that $\emptyset \notin A$, as the set containing no elements is not in the set $A$, but $\emptyset \in B$ is true, right?

In the midst of forming this question I thought of a way of looking at this question, but I am not sure whether it is correct:

So continuing with the comparison of looking at a set being a 'box', $\emptyset \subset A$ simply means that the the set $A$ has some 'free space' in the box, and so the set could be visualised by $A=\{1,2,3,\qquad\}$?, and every set\'box' has some 'empty space', whereby the 'empty box' represents simply the collection of that void.

Does this analogy capture the true picture of Why $\{1,2,3\} \neq \{1,2,3,\emptyset \}$ and $\emptyset \subset A$?

Just a very quick question,

Is the power set of $C=\{1,\emptyset\}$,

$$\mathcal{P}(C)=\{\{1\},\{ \emptyset \},\{1, \emptyset \},\emptyset \}?$$

Answer 1

Don't take the box analogy too seriously. It captures some of the aspects of the mathematical concept of a set, but not all of them. For example, we can easily imagine two boxes that are both empty, but set theory nevertheless holds that there is one and only one empty set.

Boxes have some value in explaining how $A \in B$ and $A \subseteq B$ are two different claims (which are unfortunately both expressed as "$B$ contains $A$" in common mathematical English, so one has to guess which is meant from the context). Beyond that, their value diminishes somewhat.

The real technical reality of sets is simply that they are something that either are, or are not, in the $\in$ relation to each other, such that they satisfy the Axiom of Extensionality:

A set is fully given by what its elements are. More precisely, whenever $A$ and $B$ are different sets, there must be either an $x$ such that $x\in A$ and $x\notin B$, or an $y$ such that $y\notin A$ and $y\in B$, or possibly both. If such an $x$ or $y$ does not exist, then it can only be because $A$ and $B$ are the same set.

Viewed like this, what your $A$ and $B$ are is simply the totality of answers to "is this an element of $A$?" and "is this an element of $B$?".

Because the answer to "is $\varnothing$ an element of $A$?" is different from the answer to "is $\varnothing$ an element of $B$?", the two sets are different.