Set Theory

Intersection

(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.

∩

intersection

U

the universal set

• The Intersection of two sets A and B is denoted by ∩:
  {a, b} ∩ {b, c} = {b}

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)