Q:

are elements in a subset also elements of the original set

for example  set A contains { {z} , b, c}  is "z" an element of A. 
I have an idea that it is not but want to make sure.

A:

If $B$ is a subset of $A$, and $b$ is an element of $B$, then $b$ is an element if $A$. This is by definition: "$B$ is a subset of $A$" means exactly "every element of $B$ is an element of $A$."
But that's not the situation you're describing! $\{z\}$ is not a subset of your $A$, but an element of your $A$; these aren't the same thing. And "is an element of" isn't transitive: in your example $z$ is not, in fact, an element of $A$.

