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Let $A, B$ and $C$ be three sets. If $A ∈ B$ and $B ⊂ C$, is it true that $A ⊂ C?$. If not, give an example.

This question is from my textbook. And the answer is:

No. Let $A = \{1\}, B = \{\{1\}, 2\}$ and $C = \{\{1\}, 2, 3\}.$ Here $A ∈ B$ as $A = {1}$ and $B ⊂ C.$ But $A ⊄ C$ as $1 ∈ A$ and $1 ∉ C.$ Note that an element of a set can never be a subset of itself.

I am having trouble in in understanding "But $A ⊄ C$ as $1 ∈ A$ and $1 ∉ C."$ How could $1 ∉ C?$ Clearly $C$ contains $B$ and $A$ is an element of $B$ and $A$ have $1$ so $C$ must have $1$. I would be very grateful If you answer this.

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    $\begingroup$ The statement "Note that an element of a set can never be a subset of itself" is wrong. An element of a set can be a set, and a set is always a subset of itself. $\endgroup$ – Robert Israel Oct 13 at 5:11
  • $\begingroup$ An example where $A \in B$ and $B \subset C$ and $A \subset C$ is $A = \{1\}$, $B = \{\{1\}\}$, $C = \{1, \{1\}\}$. $\endgroup$ – Robert Israel Oct 13 at 5:18
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    $\begingroup$ @RobertIsrael: …unless the book is using "subset" in the sense of "proper subset", which some still do. $\endgroup$ – Ilmari Karonen Oct 13 at 12:33
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    $\begingroup$ I understood the question to mean "Is it necessarily true that $A\subset C$?" The single counter-example given in the book is enough to answer this question in the negative. There is no mistake in the book. $\endgroup$ – TonyK Oct 13 at 14:19
  • $\begingroup$ @TonyK I was not talking about that. The question is correct so is the answer but in last phrase the book says "Note that an element ....." which seems wrong. But as Ilmari Karonen mentioned that this right if book is using subset in sense of proper subset. And he is indeed right. My book is using subset as proper set. $\endgroup$ – Shekhar Oct 13 at 15:23
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This is becuase there is a distinction between $1$ and $\{1\}$. The former is the number one. The latter is a set containing the number one. If some set $C$ contains another set, let's call it set $E$, we do not look at the members of $E$ when we consider the members of $C$. We would say the set $E$ is a member of $C$, but this does not necessarily mean that anything contained by $E$ is also in $C$.

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Note the difference of $\{1\}\in C$ (that is true) and $\{1\}\subset C$ (that is not true). The set $A=\{1\}$ is an element of $C$, not a subset. For example, $\{2\},\{3\}$ and $\{\{1\}\}$ are subsets of $C$, but $2,3$ and $\{1\}$ are not subsets of $C$, but its elements.

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