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Starting my self-study of the book "A Concise Introduction to Pure Mathematics", I came up with the following stone in the road (on page 2 incidentally).

We say T is a subset of a set S if every element of T also belongs to S... We write T ⊆ S if T is a subset of S, and T ⊄ S if not. For example, S={1, {2}, cat}

{cat} ⊆ S, {{2}} ⊆ S, {2} ⊄ S

My question is why is {2} ⊄ s(or {{2}}⊆ S) ? Why does the inclusion property not propagate down the subsets? Is this just a stricter definition of inclusion? I still wonder how I haven't thought about this question after years of university studies.

juanjo12x
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    The elements of ${ 2 }$ include $2$ and $2$ is not an element of $S$. There is no "propagating down the subsets", each set is its own object. – Ian Sep 03 '21 at 13:27
  • See also this post – Mauro ALLEGRANZA Sep 03 '21 at 13:30
  • There is a key difference between the relation "to be an element of" ($\in$) and the relation "to be a subset of" ($\subseteq$). The second one is defined in terms of the first one: "a is a subset of b iff (for every x, if x is in a, then x is in b). – Mauro ALLEGRANZA Sep 03 '21 at 13:37
  • And see also this one – Mauro ALLEGRANZA Sep 03 '21 at 13:40
  • If you apply the def of subset, you can easily check that it does not hold, because set ${ 2 }$ has the number $2$ as element and $2$ is not an element of $S$ ($2$ it is not listed in the explicit def of $S$). You must be aware of the fact (see linked posts) that the object $2$ and the set ${ 2 }$ having the object $2$ as single element are not the same. – Mauro ALLEGRANZA Sep 03 '21 at 13:51
  • I see!! So I mixed the definitions of ∈ and ⊆ . Thank you very much !!! – juanjo12x Sep 03 '21 at 13:55

2 Answers2

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The elements of the set $\{1,\{2\}, \mathrm{cat}\}$ are $1$, $\{2\}$ and $\mathrm{cat}$.

So in relation to the set $S$, we have that $\{2\}$ is an element of this set (which is a set containing 2). But it is not a subset of $S$.

This would be $\{\{2\}\}$. A set containing this element.

When we have $A\subseteq B$ that means that every element $a\in A$ has to be an element of $B$. But $\{2\}\subseteq S$ would mean that $2\in S$. Which is not the case.

Cornman
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Any set is a subset of itself. And $\{x\}$ is just an element of $\{\{x\}\}$.

sea yellow
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