Skip to main content
  1. Life
  2. Education & Schools
  3. General Education

Complements in set theory

See also

Let's look at complements in set theory. The "relative complement" is two sets is the difference between the two. As in ordinary arithmetic, we represent the difference between two sets with the minus sign. A - B is the difference between sets A and B. In set theory, we would write A - B = {x ∈ A: x ∈' B}. We can also represent the relative complement of any two sets by writing A \ B. We can thus represent the complement with the apostrophe. We would describe the symbols in the set builder notation as stating that "x is an element of A is such that x is not a member of B." It's in this sense that the relative complement of one set is precisely what is not in the other set.

De Morgan's laws apply to the relation of complements to both unions and intersections.

Thus:

(A ∪ B)' = A' ∩ B' is true as well as (A ∩ B)' = A' ∪ B'

Halmos also speaks of the "symmetric difference," also known as the "Boolean sum" of two sets. The symmetric difference or Boolean sum of two sets is the union of the difference and the inverse of that difference of two sets. Thus, we can represent it as follows:

A + B = (A - B) ∪ (B - A)

In addition to writing A + B, we can represent the symmetric difference or Boolean sum of any two sets by writing A ∆ B

Halmos, Paul. "Naive Set Theory." Litton Educational Publishing, 1960.

Advertisement

Life

  • Derek Hough
    Derek Hough brings quadruple threat talent to 'DWTS' and beyond
    Today's Buzz
  • Hookah smoking
    Young adults believe hookah smoking pose no health threat
    Camera
    7 Photos
  • Top outdoor activities
    Don't spend your summer indoors: Top outdoor activities to do with your significant other
    Camera
    10 Photos
  • Baby shower idea for men
    A new twist on baby showers is throwing a Daddy Baby-Q
    Camera
    7 Photos
  • Wedding special
    'Curvy Brides' offers a new look into every bride's pursuit for her picture perfect wedding gown
    Camera
    7 Photos
  • Morbid obesity
    Health: Morbid obesity decreases life span by up to 16 years
    Camera
    7 Photos