Members of sets related to subsets

Question

I was recently reading about sets and read that $B$ is a subset of $A$ when each member of $B$ is a member of $A$. However, I am not sure about whether this requires the members of $A$ to simply be members of $B$, or if they could be part of $A$ in some other way - i.e. embedded within a set inside $A$.

I tried to think about the following example:

$$X = \{10,\{x\}\}$$ $$Y = \{x\}$$

Does this mean that $Y$ is not a subset of $X$, as $x$ is a member of $Y$, but $x$ is not a member of $X$? If this is the case, I think I could say that $Z = \{\{x\}\}$ is a subset of $X$.

Or, is $Y$ a subset of $X$ as "x" exists somewhere within $X$, even though it is an element of a set, which itself is an element of $X$? I find this unlikely but cannot get past this idea. Thank you.

Answer 1

"I am not sure about whether this requires the members of $A$ to simply be members of $B$, or if they could be part of $A$ in some other way - i.e. embedded within a set inside $A$."

Former.

"Does this mean that $Y$ is not a subset of $X$, as $x$ is a member of $Y$, but $x$ is not a member of $X$? If this is the case, I think I could say that $Z = \{\{x\}\}$ is a subset of $X$."

Yes and yes.

"Or, is $Y$ a subset of $X$ as "x" exists somewhere within $X$, even though it is an element of a set, which itself is an element of $X$?"

No.

You can draw a Venn diagram to get a better idea. So, for example, a disk with centre at origin and radius $1$ is a subset of a disk with centre at origin and radius $2.$ For $A$ to be a subset of $B$ it is merely sufficient that $A$ sits inside $B.$ There is no restriction on $B.$

Answer 2

In general you can't consider $x$ and $\{x\}$ as the same thing, so $Y\subseteq X$ means $$\forall y\in Y(y\in X)$$