Let $A$, $B$ and $C$ be three sets. If $A$ belongs to $B$ and $B$ is a subset of $C$, is it true that $A$ is a subset of $C$ too?

Let $A$, $B$ and $C$ be three sets. If $A$ belongs to $B$ and $B$ is a subset of $C$, is it true that $A$ is a subset of $C$ too?

No. Let $A=\{1\}$, $B=\{\{1\},2\}$ and $C=\{\{1\},2,3\}$. Here $A$ belongs to $B$ as $A=\{1\}$ and $B$ is a subset of $C$. But $A$ is not a subset of $C$ as $1$ belongs to $A$ and $1$ doesn't belong to $C$. Note that an element of a set can never be a subset of itself.

I am confused by the last note. Is the above mentioned explanation correct? Can someone explain in a better way?

• Hello and welcome! The statement "If $A$ is a subset of $B$ and $B$ is a subset of $C$, then $A$ is a subset of $C$" is correct. But the statement "If $A$ is an element of $B$ and $B$ is a subset of $C$, then $A$ is a subset of $C$" is not correct. The textbook gives an instance for the second statement. Try to make an instance for the first statement! – Hans Engler Dec 2 '14 at 3:53

That last line seems completely unclear, and probably false, too. For example $\varnothing$ is both an element and a subset of $\{\varnothing\}$. And $\{1\}$ is an element and a subset of $\{1,\{1\}\}$.
But in the example, this is indeed the case, that $A\in B$, but $A\nsubseteq C$. Since $1$ is the unique element of $A$, but $1\notin C$.