Question

Determine whether the symmetric difference is associative; that is, if A, B and C are sets, does it follows that \(A \oplus (B \oplus C) = (A \oplus B) \oplus C\)

Short answer

The symmetric difference is associative.

Step-by-step solution

Step: 1

We know that,

\(A \oplus (B \oplus C) = (A - (B - C) \cup (C - B)) \cup ((B - C) \cup (C - B) - A)\)

Step: 2

Also, we know that,

\(((A \oplus B) \oplus C = ((A - B) \cup (B - A) - C) \cup (C - (A - B) \cup (B - A))\)

Therefore we get, \(A \oplus (B \oplus C) = (A \oplus B) \oplus C\)