1
$\begingroup$

I was solving a list of exercises from homework and I answered one question wrong because I thought that a singleton set and its element were the same thing...

I found only a few results from a Google search:

https://www.google.com/#q=%22set+is+not+the+same+as+its+element%22

May anyone please explain this to me?

$\endgroup$
3
  • $\begingroup$ Just as an example: $\{x\}$ has exactly one element, namely $x$. However, how many elements $x$ has depends on $x$. $\endgroup$
    – Carsten S
    Mar 23, 2014 at 15:16
  • $\begingroup$ @Git Gud: Oh, sorry, the problem is that I often receive more than a good answer, I normally just UP vote them, and let the votes of other user decide which answer is the best (by receiving more votes than the others). But I will try to use the accept answer feature from now on. $\endgroup$
    – João Rimu
    Mar 23, 2014 at 15:20
  • $\begingroup$ @JoãoRimu That's a sensible approach. Do note, however, that I didn't tell you to accept any answer or anything, it's your choice. But I do recommend you accept some answers. $\endgroup$
    – Git Gud
    Mar 23, 2014 at 15:21

3 Answers 3

Reset to default
5
$\begingroup$

First of all, it can't always be the same. $\{\varnothing\}\neq\varnothing$. On the right-hand side, we have the empty set which has no elements but on the left-hand side we have a set which does include an element.

More generally, one of the axioms of modern set theory allows us to prove that no set is an element of itself. In particular $\{x\}\notin\{x\}$. Since the only element of $\{x\}$ is $x$, when we say $\{x\}\notin\{x\}$ we actually say that $x\neq\{x\}$.

$\endgroup$
4
$\begingroup$

If a singleton set and its element were the same thing, then the set $\{\{1,2,3\}\}$, which has just one member, would be the same thing as the set $\{1,2,3\}$, which has three members. Suppose one wants to enumerate the partitions of the set $\{1,2,3\}$. They are: \begin{align} & \{\{1,2,3\}\} \\[6pt] & \{\{1,2\},\ \ \{3\}\} \\[6pt] & \{\{1,3\},\ \ \{2\}\} \\[6pt] & \{\{2,3\},\ \ \{1\}\} \\[6pt] & \{\{1\},\ \ \{2\}, \ \ \{3\}\} \end{align}

The first partition listed has one part, not three. That part has three elements, not one.

$\endgroup$
2
$\begingroup$

A bag containing an orange is not the same thing as the orange itself.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.