# When does X \ A = $\emptyset$?

Where A is any set, what set X would fulfill the equation $X \setminus A = \emptyset$? An empty set has no elements, so what I would guess is that X would have to be the same set as A, since removing all real elements of X would leave you with an empty set. Does that make sense?

Additionally, is $X \triangle A = \emptyset$ (symmetric difference) the same way? Wouldn't they have to be the same set for that expression to return an empty set?

You answer the first question incorrectly. The answer should be any subset of A. The symmetric difference however is only the empty set if and only if $X=A$.

You could write it out by looking at what happens if there are elements of X that are not an element of A and the other way around.

• I should have wrote subset, yes. I'm glad to see my line of thinking is correct for this one, thanks. – SolidSnackDrive Jun 13 '18 at 22:56
• You are welcome – Stan Tendijck Jun 13 '18 at 22:57

$X\setminus A=\varnothing$ if and only if $X\subseteq A$.

There results that $X\mathbin\Delta A=(X\setminus A)\cup(A\setminus X)$ is empty if and only if each of $X\setminus A$ and $A\setminus X$ is, i.e. if and only if $X\subseteq A$ and $A\subseteq X$, which indeed means $X=A$.

An empty set has no elements, so what I would guess is that X would have to be the same set as A, since removing all real elements of X would leave you with an empty set. Does that make sense?

Yes, it makes sense. In removing the elements of $A$ you end up removing all the elements of $X$, so all the elements of $X$ must be elements of $A$ and removed.

But that just tells you the elements of $X$ are in $A$. It doesn't tell you that anything about the elements of $A$ that may or may not be in $X$.

So that means $X \subset A$. But it doesn't mean $A = X$. It is possible that $X \subsetneq A$.

...

As for symmetric difference, if I understand it correctly, is $(X\setminus A) \cup (A\setminus X) = (X\cup A)\setminus (X\cap A)$ is the set of all elements in $X$ or in $A$ but not both, right?

In that case ....$X \Delta Y=\emptyset$ means There are no elements in $X$ that aren't in $A$ and there aren't any elements in $A$ that aren't in $X$ so.... $X$ and $A$ have the same elements.

So $X = A$.