Set Symmetric Difference Operation
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A Set Symmetric Difference Operation is a n-set operation that produces the set of all set members in one of [math]\displaystyle{ A or B }[/math] but not both sets.
- AKA: Disjunctive Union.
- Context:
- It can corresponds to the XOR Operation in Boolean Logic.
- It can be represented as [math]\displaystyle{ A \ominus B }[/math], [math]\displaystyle{ A\,\triangle \,B }[/math], or [math]\displaystyle{ A \oplus B }[/math].
- …
- Counter-Example(s):
- See: Set Operation, Set Difference.
References
2017
- (Wikipedia, 2017) ⇒ https://en.wikipedia.org/wiki/symmetric_difference Retrieved:2017-6-2.
- In mathematics, the symmetric difference, also known as the disjunctive union, of two sets is the set of elements which are in either of the sets and not in their intersection. The symmetric difference of the sets A and B is commonly denoted by : [math]\displaystyle{ A\,\triangle\,B, }[/math] or : [math]\displaystyle{ A \ominus B, }[/math] or : [math]\displaystyle{ A \oplus B. }[/math] For example, the symmetric difference of the sets [math]\displaystyle{ \{1,2,3\} }[/math] and [math]\displaystyle{ \{3,4\} }[/math] is [math]\displaystyle{ \{1,2,4\} }[/math] .
The power set of any set becomes an abelian group under the operation of symmetric difference, with the empty set as the neutral element of the group and every element in this group being its own inverse. The power set of any set becomes a Boolean ring with symmetric difference as the addition of the ring and intersection as the multiplication of the ring.
- In mathematics, the symmetric difference, also known as the disjunctive union, of two sets is the set of elements which are in either of the sets and not in their intersection. The symmetric difference of the sets A and B is commonly denoted by : [math]\displaystyle{ A\,\triangle\,B, }[/math] or : [math]\displaystyle{ A \ominus B, }[/math] or : [math]\displaystyle{ A \oplus B. }[/math] For example, the symmetric difference of the sets [math]\displaystyle{ \{1,2,3\} }[/math] and [math]\displaystyle{ \{3,4\} }[/math] is [math]\displaystyle{ \{1,2,4\} }[/math] .
2017
2017
- https://en.wikipedia.org/wiki/Set_(mathematics)#Complements
- QUOTE: An extension of the complement is the symmetric difference, defined for sets A, B as :[math]\displaystyle{ A\,\Delta\,B = (A \setminus B) \cup (B \setminus A). }[/math] For example, the symmetric difference of {7,8,9,10} and {9,10,11,12} is the set {7,8,11,12}. The power set of any set becomes a Boolean ring with symmetric difference as the addition of the ring (with the empty set as neutral element) and intersection as the multiplication of the ring.