Set Theory
Symmetric Difference
Term
- symmetric difference, n.
- (sets) The symmetric difference of sets A and B
is the set of all elements of A or B
which are not in both A and B.
Symbolically,
A Δ B = {x|(x ∈ A ∧ x
∉ B) ∨ (x ∉ A ∧ x ∈ B)}
Symbol
- Δ
- (sets) symmetric difference
Notation
- symmetric difference
- The symmetric difference of two sets A and B i
s denoted by Δ,
e.g.,
{a, b} Δ {b, c} = {a, c}
Venn Diagram
Closure
- symmetric difference
- =`The set of sets is closed under symmetric difference.
That is, the symmetric difference of two sets is a set.
Associativity
- symmetric difference
- Difference of sets is associative:
that is, for all sets A, B, and C,
(A Δ B) Δ C = A Δ (B Δ C)
Commutativity
- symmetric difference
- Symmetric difference of sets is commutative:
that is, for all sets A and B,
A Δ B = B Δ A
Identities
- symmetric difference
- The empty set, ∅, is a right identity for set difference.
that is, for any set A,
A Δ &empty = A
- The empty set, ∅, is a leftt identity for set difference.
that is, for any set A,
&empty Δ A = A
Idempotency
- symmetric difference
- Sets are not idempotent under symmetric difference:
that is, in general, for set A,
A Δ A = ∅ ≠ A
Notation
- symmetric difference
- The symmetric difference of two sets A and B i
s denoted by Δ,
e.g.,
{a, b} Δ {b, c} = {a, c}
Remarks
- Symmetric difference is a binary operation.
- The symmetric difference is conceptually similar to the
exclusive or operation.
Reference Links
External Links
