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The Associativity of the Symmetric Difference

The symmetric difference of two sets A and B is defined by A AB = (A \ B) U (B \ A). It is easy to verify that A is commutative. However, associativity of A is not as straightforward to establish, and usually it is given as a challenging exercise to students learning set operations (see [1, p. 32, exercise 15], [3, p. 34, exercise 2(a)], and [2, p. 18]). In this note we provide a short proof of the associativity of A. This proof is not new. A slightly different version appears in Yousefnia [4]. However, the proof is not readily accessible to anyone unfamiliar with Persian. Consider three sets A, B, and C. We define our universe to be X = A U ? U C. For any subset U of X, define the characteristic function of U by