(set theory) 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
The set which, in a certain context, contains everything of
interest.
idempotent
A value is idempotent under a binary operation if the result of
applying that operation to the value is the same value.
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.
Symbols
∩
intersection
U
the universal set
Notation
The Intersection of two sets A and B is denoted by ∩: {a, b} ∩ {b, c} = {b}
Venn Diagram
Properties
Closure
The set of sets is closed under intersection.
That is, the intersection of two sets is a set.
Associativity
Intersection of sets is associative: that is, (A ∩ B) ∩ C = A ∩ (B ∩
C)
Commutativity
Intersection of sets is commutative: that is, A ∩ B = B ∩ A
Identity
The universal set, U, is
both a right and left identity for set intersection: that is, for
all sets A, A ∩U=
A U∩ A = A
Idempotency
Sets are idempotent under intersection: that is, for all sets
A, A ∩ A = A
Distributivity
Intersection distributes over union: that is, A ∩ (B ∪ C) = (A ∪ B) ∩
(A ∪ C)