Is A a subset or an element of set C? Let A, B, C be 3 sets. If A belongs to B, and B is a subset of C, is it true that A is a subset of C?
They say A is a set. So A should be a subset of C. 
But my textbook says it’s not because A is an element (as seen in the belongs to area). 
So what is the question trying to say? I’m really confused. 
 A: Note: In general, an object can be both a set on its own, and an element of another set.
In your particular case, as J. W. Tanner's question comment indicates, $A$ is an element of $C$ (due to it being an element of $B$ and $B$ being a subset of $C$) regardless of its status of whether or not it is a set on its own.
For $A$ to be a subset of $C$, it must either be empty or contain at least one element of $C$. However, the only specific element of $C$ you are given is $A$ itself, but $A$ doesn't contain itself as an element of its own set.
A: The question tests if you've understood the difference between membership of a set and subsets. A quick example to help settle this. 
Let $A=\{1,2\},B=\{3,4\}$ and $C=\{\{1,2\},3,4\}$. 
The three objects $\{1,2\}$, $3$ and $4$ are elements of $C$. So, the object $A$ (which is a set on its own) is an element of $C$.
Given two sets $S$, $T$:
$T$ is subset of $S$, if and only if $x \in T \implies x \in S \hspace{3mm} \forall x$. 
Thus, $B=\{3,4\}$ is a subset of $C$.
Further, the singleton set that contains the object $A$:
$T = \{A\} =\{\{1,2\}\}$ is a subset of $C$. 
A: *

*In general, an element of a subset of a set S is not a subset of S. 


That is, from " it is an element of a subset of S " you cannot infer automatically that it is also a subset of S. The reason is not a sufficient one. 


*

*But note also, that it is not either a sufficient reason to conclude that it is not a subset of S. 


It can happen that an element of a subset of S is also a subset of S. 


*

*Example :


Since $2=_{df}\{0,1\}$, it is true that $1\in 2$. 
Since $3 =_{df} \{0,1,2\}$,   it is true that $2\subseteq 3$ ( that is : 2 is a subset of 3). 
Since $1=_{df}\{0\}$ , $1\subseteq 3$. 
So, number $1$ is an element of a subset of number$3$ and is also a subset of $3$. 
Note : logically speaking , the key point is to understand this : the fact that " A implies B" is not a true general rule does not mean that "A implies not-B" is a true general rule. 
