## Symmetric difference

Determine which of the following statements about the symmetric difference $$\oplus$$ defined as $$A\oplus B= (A\cup B)\setminus (A\cap B)$$ are true, and which are false.

• $$A\oplus B= (A\cap \overline B)\cup (\overline A\cap B)$$
• $$A\oplus B= B\oplus A$$
• $$A\oplus (B \oplus C)= (A \oplus B) \oplus C$$
• $$A\oplus(B \oplus A)= A$$
• $$A\oplus A= \emptyset$$
• $$A\oplus \emptyset= A$$

If a statement is false, correct it using the smallest possible change if possible.

• #### Solution

For the first, third and fourth statements it's helpful to draw a Venn diagram and mark the sets corresponding to the expressions on the left and right.

It's possible to prove some of the statements formally, e.g. by transforming set expressions, e.g.

$$A\oplus B= (A\cup B)\setminus (A\cap B) = (B\cup A)\setminus (B\cap A) = B\oplus A$$. Here we have used the fact that $$\cap$$ and $$\cup$$ are commutative.

$$A\oplus A= (A\cup A)\setminus (A\cap A) = A \setminus A = \emptyset$$

$$A\oplus \emptyset= (A\cup \emptyset)\setminus (A\cap \emptyset) = A \setminus \emptyset = A$$

We can refute the fourth statement formally using the second, third and fifth:
$$A\oplus(B \oplus A)= A\oplus(A \oplus B)= (A\oplus A) \oplus B =\emptyset \oplus B = B \oplus \emptyset= B$$

$$A\oplus(B \oplus A)= A$$ is invalid; the statement $$A\oplus(B \oplus A)= B$$ e.g. is true.  