# 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?

## 3 Answers

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 $X = A$.