Removing elements from an empty set

Question

If I try to remove elements from an empty set, would I get an empty set or would this operation be undefined?

Answer 1

Obviously you cannot actually remove elements from an empty set, so I'm guessing that your question is intended to refer to the operator $\setminus$, as in $\emptyset\setminus\{1,2\}$. If this is so, the question pertains not only to the empty set, but to any case in which the left-hand operand contains elements that the right-hand operand doesn't contain. The operator is defined such that these elements are irrelevant, and only the elements actually contained in the left-hand operand are actually removed; so the operation is well-defined and yields $\emptyset\setminus\{1,2\}=\emptyset$.

Answer 2

A definition of the set $A\setminus B$ (read "A setminus B") is $$ A\setminus B=\{x\,|\,x\in A,x\notin B\}. $$ In particular, $A\setminus B\subseteq A$. If $A=\emptyset$, this proves that $A\setminus B=\emptyset$ for every set $B$.