Symmetric difference
Task number: 3343
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:
\(A\oplus(B \oplus A)= A\oplus(A \oplus B)= (A\oplus A) \oplus B =\emptyset \oplus B = B \oplus \emptyset= B\)Answer
\(A\oplus(B \oplus A)= A\) is invalid; the statement \(A\oplus(B \oplus A)= B\) e.g. is true.
