Logic - define the empty set

Question

I have the signature $\sum=(\setminus)$, while $M=P(\mathbb{N})$.

I want to define Empty($S$)= True iff $S=\emptyset$.

Is this legit: Empty($S$)=$\forall A((A\setminus S)=A)$ ?

I know that this one works: Empty(S)=$(S=S \setminus S)$, but I want to understand if the first one is also correct.

Thanks!

Answer 1

Technically you haven't fully specified the structure you're looking at: you've told us its underlying set but not its interpretation of the symbol "$\setminus$" in the language. Of course it's clear notationally what it should be, namely set difference, but this is worth observing.

Yes, that works. $X\setminus Y$ is the set of elements of $X$ which aren't in $Y$, so clearly we have $X\setminus\emptyset=X$ for all $X$. Meanwhile, if $Y\not=\emptyset$ then - picking some $a\in Y$ - we have $\{a\}\setminus Y=\emptyset\not=\{a\}$. So your property does indeed characterize $\emptyset$ exactly.

Another example which works is "$\forall X(S\setminus X=S)$."