Set Theory
Set Difference
Terms
- set difference, n.
- (sets) The difference of sets A and B is
the set of all elements of A
which are not also elements of B.
Symbolically,
A - B = {x|x ∈ A ∧ x ¬in B}
- relative complement, n.
- (math) See set difference.
Symbols
- -
- (sets) set difference
- \
- (sets) set difference
Venn Diagram
Closure
- set difference
- The set of sets is closed under set difference.
That is, the difference of two sets is a set.
Associativity
- set difference
- Difference of sets is not associative:
that is, in general, for sets A, B, and C,
(A - B) - C ≠ A - (B - C)
Commutativity
- set difference
- Difference of sets is not Commutative:
that is, in general for sets A and B,
A - B ≠ B - A
Identities
- set difference
- The empty set, ∅, is a right identity for set difference.
that is, in general, for set A,
A - &empty = A
- There is no left identity for set difference.
Idempotency
- set difference
- Sets are not idempotent under difference:
that is, in general, for set A,
A - A = ∅ ≠ A
Terms
- set difference, n.
- (sets) The difference of sets A and B is
the set of all elements of A
which are not also elements of B.
Symbolically,
A - B = {x|x ∈ A ∧ x ¬in B}
- relative complement, n.
- (math) See set difference.
Notation
- The difference of two sets A and B is denoted by - or \:
{a, b} - {b, c} = {a}
{a, b} \ {b, c} = {a}
Remarks
- Difference is a binary operation.
- The difference of sets A and B, A - B, is also referred to as
the relative complement of B in A.
Reference Links
External Links
