Set theory bracket notation, what is excluded $X=\{\emptyset,\{\emptyset\},\{\{\emptyset\}\}\}$ and $Y=X\setminus\{\{\emptyset\}\}$

Question

If $X=\{\emptyset,\{\emptyset\},\{\{\emptyset\}\}\}$ and $Y=X\setminus\{\{\emptyset\}\}$

then what element is excluded from $X$? Is it $\{\{\emptyset\}\}$, or $\{\emptyset\}$?

In a similar vein, if $Z=\{a, b, c\}$, does it make sense to say $Z\setminus a$?

Thanks

Answer 1

For your first question, if $A$ and $B$ are sets, then $A \setminus B := \{ x \mid x \in A \wedge x \not \in B\}$. Thus, $X \setminus \{\{\emptyset\}\} = \{\emptyset,\{\{\emptyset\}\}\}$.

For the second question, if you follow the previous definition strictly, then it doesn't make sense to write $Z \setminus a$ although it might be allowed via convention given how clunky $Z \setminus \{a\}$ looks (especially in view of the example you provided).

Answer 2

$$X=\{\emptyset,\{\emptyset\},\{\{\emptyset\}\}\}$$

and $$ Y=X\setminus\{\{\emptyset\}\}= \{\emptyset,\{\{\emptyset\}\}\} $$ because $\{\emptyset\}$ is removed from your $X$.

For your next question regarding $$ Z=\{a, b, c\}$$ $Z\setminus a$ does not make sense unless $a$ is a set.