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## Symmetric difference

Definition 3.2.29   Let and be any two sets. The symmetric difference of and is the set denoted by and who consists of all the elements that belong to exactly one of the sets and , and not to both, i.e.  We can rewrite the definition a follows: (exclusive) or Proposition 3.2.30   Let and be any two sets. Then Proposition 3.2.31 (the commutative law)   Let and be two sets. Then the following equality holds: Proposition 3.2.32 (the associative law)   Let , and be three sets. Then the following equality holds:  Proof. We can write down a proof using truth tables; as this is exactly the proof of the associative law of the exclusive or'' from Chapter 2, we are done. Anyway, we wish at least to outline a proof using properties of operations on sets. On the one hand, we have: (the reader will fill the missing rows):        On the other hand, we have:        Whence the required equality. A much more simple proof can be written, using characteristic functions (v.i. prop associativity of symmetric difference 2).    Next: The cartesian product of Up: Operations with sets Previous: Difference   Contents
root 2002-06-10