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symmetric difference∗
CWoo†
2013-03-21 12:51:09
The symmetric difference between two sets A and B, written A4B, is the
set of all x such that either x ∈ A or x ∈ B but not both. In other words,
A4B := (A ∪ B) \ (A ∩ B).
The Venn diagram for the symmetric difference of two sets A, B, represented
by the two discs, is illustrated below, in light red:
.
A
B
.
Properties
Suppose that A, B, and C are sets.
• A4B = (A \ B) ∪ (B \ A).
• A4B = Ac 4B c , where the superscript c denotes taking complements.
• Note that for any set A, the symmetric difference satisfies A4A = ∅ and
A4∅ = A.
• The symmetric difference operator is commutative since A4B = (A\B)∪
(B \ A) = (B \ A) ∪ (A \ B) = B4A.
∗ hSymmetricDifferencei created: h2013-03-21i by: hCWooi version: h30916i Privacy
setting: h1i hDefinitioni h03E20i
† This text is available under the Creative Commons Attribution/Share-Alike License 3.0.
You can reuse this document or portions thereof only if you do so under terms that are
compatible with the CC-BY-SA license.
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• The symmetric difference operation is associative: (A4B)4C = A4(B4C).
This means that we may drop the parentheses without any ambiguity, and
we can talk about the symmetric difference of multiple sets.
• Let A1 , . . . , An be sets. The symmetric difference of these sets is written
n
4 Ai .
i=1
In general, an element will be in the symmetric difference of several sets
iff it is in an odd number of the sets.
It is worth noting that these properties show that the symmetric difference
operation can be used as a group law to define an abelian group on the power
set of some fixed set.
Finally, we note that intersection distributes over the symmetric difference
operator:
A ∩ (B4C) = (A ∩ B)4(A ∩ C),
giving us that the power set of a given fixed set can be made into a Boolean
ring using symmetric difference as addition, and intersection as multiplication.
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