properties of symmetric difference
If , then , because and .
, because , and .
, because and .
(hence the name symmetric difference).
, because .
(distributivity of over ) .
, which is , one of the properties of set difference (see proof here (http://planetmath.org/PropertiesOfSetDifference)). This in turns is equal to . ∎
(associativity of ) .
Let be a set containing as subsets (take if necessary). For a given , let be a function defined by . Associativity of is then then same as showing that , since .
By expanding , we have
It is now easy to see that the last expression does not change if one exchanges and . Hence, and this shows that is associative. ∎
Remark. All of the properties of on sets can be generalized to (http://planetmath.org/DerivedBooleanOperations) on Boolean algebras.
|Title||properties of symmetric difference|
|Date of creation||2013-03-22 14:36:56|
|Last modified on||2013-03-22 14:36:56|
|Last modified by||CWoo (3771)|