n. (sets) The intersection of sets A and B is the set of
all elements of A which are also elements of B.
Symbolically,
A ∩ B = {x:x ∈ A ∧ x ∈ B}
universal set
n. (sets) The set which, in a certain context, contains everything of interest.
idempotent
n. (sets) A value is idempotent
under a binary operation if the result of
applying that operation
to the value is the same value.
Symbol
∩
(sets) intersection
Notation
intersection
The intersection of two sets A and B is denoted by
A ∩ B
which is read
A intersection B
Venn Diagram
Closure
intersection
The set of sets is closed under intersection.
I.e. intersection always has a set value.
Symbolically,
∀ A, B ∈ ℘, (A ∩ B) ∈ ℘
Associativity
intersection
For all sets, intersection is associative.
That is, the result does not depend on the order of evaluation.
Symbolically,
∀A, B, C ∈ ℘, ((A ∩ B) ∩ C) = (A ∩ (B ∩ C))
Commutativity
intersection
For all sets, intersection is commutative.
That is, the result does not depend on the order of the operands.
Symbolically,
∀A, B ∈ ℘, (A ∩ B) = (B ∩ A)
Idempotency
intersection
For all sets, intersection is idempotent.
That is, the intersection of any set with itself is the same value.
Symbolically,
∀A ∈ ℘, (A ∩ A) = A
Identity
intersection
The universal set, U, is a left identity for intersection of sets. Symbolically,
∀ A ∈ ℘, U ∩ A = A
The universal set, U, is a right identity for intersection of sets. Symbolically,
∀ A ∈ ℘, A ∩ U = A
Remarks
Intersection is a binary operation.
The universal set is also referred to as the universe.
Commonly, each investigation has its own universal set,
which includes everything under investigation, but not
everything in the actual known universe.
Intersection distributes over union: that is,
A ∩ (B ∪ C) = (A ∪ B) ∩ (A ∪ C)
