# 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