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In Discrete Math, it's trivial to say that $x \ne \{x\}$, why is that?

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  • $x\in{x}$, but ${x}\notin x$. – Bernard Dec 27 '20 at 17:01
  • Think of sets as trash cans that have objects in them (although the empty set -- an empty trash can -- is allowed). $x$ is an object (which might be a trash can itself, either empty or not empty), and ${x}$ is what you have when you put $x$ in a trash can. – Dave L. Renfro Dec 27 '20 at 17:01
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    @Dave: With analogies like these no wonder people think of set theory as a system with "garbage theorems". :-D – Asaf Karagila Dec 27 '20 at 19:16
  • @Asaf Karagila: I came up with this sometime in the mid 1980s when teaching a low level "finite matheamtics" class (the kind taken by non-science majors for their required university math elective; basically one of those math appreciation courses) and at one point the book covered some basic set theory (what union, intersection, etc. are along with simple Venn diagrams) in which things like "is ${x}$ an element of ${x,,{{x}},,{x,y}}$" were asked (actually, this is probably harder than any of the actual questions!) (continued) – Dave L. Renfro Dec 28 '20 at 08:50
  • and during the lecture I was trying to think of an analogy I could use, looked around the room, and I saw a trash can by the exit door (looked like this, except with faded dark green paint, not shinny new like this one), which I realized would be perfect, so I picked it up and talked about things in trash cans, including one or more trash cans (each with various objects in them) in a trash can. Also the distinction between putting two erasers in the trash can and ${x,x} = {x}.$ – Dave L. Renfro Dec 28 '20 at 09:00
  • actually, this is probably harder than any of the actual questions! --- In case anyone reading here missed the subtlety involved, note that if $x \neq y,$ then the answer is NO; and if $x=y,$ then the answer is YES. – Dave L. Renfro Dec 28 '20 at 11:31

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Because $x$ is an element (such as a real number $x\in \mathbb{R}$, or an integer, ...) while $\{x\}$ is a set where the only element is $x$ : it is called a singleton.

So $x\in \{x\}$ but these are not the same types of objects.

And if you take another element y, then $x\in \{x,y\}$ and $y\in \{x, y\}$.

I'm not sure the analogy will clarify it, but it's the same thing as a socket, and a socket in a drawer. The socket is the only thing in the drawer now, but it's not the same as the drawer containing the socket. And you can add other sockets in your drawer...

But be careful : $x$ can also be a set itself, for instance $x=\{2, 3\}$, then $\{x\} = \{\{2, 3\}\}$ so $\{x\}$ is a set of set...

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