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Hi I am just confused on what my teacher was saying.

So,

We had use the set definition of naturals ie

that $0= \emptyset$

$1=\{0\}=\{\emptyset\}$

$2=\{0,1\}=\{{\emptyset,\{\emptyset\}}\}$

and then he said,

$$3 \not\subseteq \{3,4\}$$

and I just don't seem to understand why.

My understanding was that $X \subseteq Y$ iff every element of X is also an element of Y.

So to try and help me understand I have written out

$3=\{0,1,2\}$

$\{3,4\}=\{\{{0,1,2\},\{0,1,2,3\}}\}$

So it the reason that it is not a subset is that for example $0=\emptyset$ is an element in the set 3 ( is set 3 contains 3 elements, 0, 1 and 2) while the RHS contains only two elements, both being sets themselves?

I think I am just a bit confused on the notation , and because it seemed counter intuitive. Is this reasoning correct? is there something I am missing with this basic question?

Thank you all

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    $\begingroup$ Notice that $3\in\{3,4\}$ but $3\not\subseteq\{3,4\}$. In other words: $3$ isnt a subset of $\{3,4\}$. $\endgroup$ – Masacroso Sep 2 '16 at 22:30
  • $\begingroup$ @Masacroso Thanks but this is what I am asking to understand $\endgroup$ – PersonaA Sep 2 '16 at 22:33
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    $\begingroup$ The subsets of $\{3,4\}$ are $\emptyset$, $\{3\}$, $\{4\}$ and $\{3,4\}$. The number $3$ is not one of these four subsets. $\endgroup$ – Masacroso Sep 2 '16 at 22:36
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You have to distinguish between elements and subsets. An ELEMENT is said to be in a set $M$, if $M$ contains it. A SET $K$ is a subset of $M$, if every element in $K$ is also in $M$.

The set corresponding to the number $3$ is not the set {$3$}, nor is it one of the $3$ other subsets of {$3,4$}.

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    $\begingroup$ Thanks, this and the comments have completely cleared it up $\endgroup$ – PersonaA Sep 2 '16 at 22:42

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