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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

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    $\begingroup$ It is the $\{\emptyset \}$ element. $\endgroup$
    – King Tut
    Mar 6 '18 at 15:11
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    $\begingroup$ This notation is for substracting a subset of your original set. Therefore, you would have the element that would be excluded from $X$ is $\{ \emptyset \}$ and the notation should be $Z\backslash \{ a \}$. $\endgroup$ Mar 6 '18 at 15:11
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    $\begingroup$ The elements inside a set "listed" in the form $X= \{ a,b,c \}$ are those inside the outer brackets, i.e. $a,b$ and $c$. Thus, in $\{ \{ \emptyset \} \}$, the (only) element is $\{ \emptyset \}$. $\endgroup$ Mar 6 '18 at 15:22
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    $\begingroup$ An illustration: $X \setminus \{\{\emptyset\}\}=\{ \color{red} {\emptyset,\{\emptyset\},\{\{\emptyset\}\}}\} \setminus \{ \color{red}{\{\emptyset\}}\}$ $\endgroup$
    – King Tut
    Mar 6 '18 at 15:23
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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).

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

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.

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