# Prove that sets $A$ and $B$ are disjoint iff $A \cup B = A \bigtriangleup B$

I'm studying for my exam and I came up with this little proof, but I'm wary because the professor took a much longer approach. Am I right in saying that a symmetric difference is the same as the difference between a union and an intersection? Thanks in advance.

Suppose $A$ and $B$ are disjoint.

This means $A \cap B = \varnothing$

Since $A \bigtriangleup B$ is the set of all elements that belong to $A$ or $B$ but not to both, $$A \bigtriangleup B = A \cup B - A \cap B$$

$A \cap B = \varnothing$, therefore $A \bigtriangleup B = A \cup B$.

• This is half of the proof. You are not done. And if the symmetric difference was defined differently (for instance, as $(A\setminus B)\cup(B\setminus A)$), then you still need to prove that it is union minus intersection. Dec 5 '14 at 6:26
• @AndresCaicedo what would be the second half?
– epon
Dec 5 '14 at 8:59

$A \Delta B=(A-B)\cup(B-A)$ $=(A\cap B^C)\cup(B\cap A^C)$ $=(A\cup B)\cap(A^C\cup B^C)$ $=(A\cup B)\cap(A\cap B)^C=(A\cup B)-(A\cap B)$

You can just see the proprieties of definition of symmetrical difference If $A\cap B= \emptyset$ then $A\backslash B = A$ and $B\backslash A=B$, then you can say that the definition of symmetrical difference is $A\bigtriangleup B = (A\backslash B) \cup (B\backslash A)$ there for the initial condition you can say $A\bigtriangleup B = A\cup B$

The thing the professor might do it longer than yours is because of the arguments, but in essence its the same, but be sure to argument every set its a big deal in mathematics.


Here is (the beginning of) a 'logical' way to prove your $$\tag 0 A \cap B = \empty \;\equiv\; A \cup B = A \diff B$$ i.e., by using the definitions to go to the element level, and using the laws of logic to simplify and complete the proof.

We will use the following not-too-well-known definition of $\;\diff\;$: for all $\;x\;$, $$\tag 1 x \in A \diff B \;\equiv\; x \in A \not\equiv x \in B$$ and we use the fact that $\;\equiv\;$ and $\;\not\equiv\;$ are symmetric, associative, and even mutually associative.

We start at the most complex side of $\ref 0$, and calculate

$$\calc A \cup B = A \diff B \calcop\equiv{set extensionality, definition of \;\cup\;, definition \ref 1 of \;\diff\;} \langle \forall x :: x \in A \lor x \in B \;\equiv\; x \in A \not\equiv x \in B \rangle \calcop\equiv{logic: extract \;\lnot\; -- to prepare for \ref 2, see below} \langle \forall x :: \lnot (x \in A \lor x \in B \;\equiv\; x \in A \;\equiv\; x \in B) \rangle \calcop\equiv{...} \endcalc$$

Now finish it yourself, by applying the 'golden rule' $$\tag 2 P \;\equiv\; Q \;\equiv\; P \lor Q \;\equiv\; P \land Q$$ and the definitions of $\;\cap\;$ and $\;\empty\;$ to reach the left hand side of $\ref 0$.