Editing Complement (set theory) (section)

{{Short description|Set of the elements not in a given subset}}
{{multiple image
| align = right
| image1 = Venn01.svg
| width1 = 150
| alt1 = A circle filled with red inside a square. The area outside the circle is unfilled. The borders of both the circle and the square are black.
| caption1 = If {{mvar|A}} is the area colored red in this image…
| image2 = Venn10.svg
| width2 = 150
| alt2 = An unfilled circle inside a square. The area inside the square not covered by the circle is filled with red. The borders of both the circle and the square are black.
| caption2 = … then the complement of {{mvar|A}} is everything else.
}}

In [[set theory]], the '''complement''' of a [[Set (mathematics)|set]] {{mvar|A}}, often denoted by <math>A^c</math> (or {{math|''A''′}}),<ref>{{Cite web|title=Complement and Set Difference|url=http://web.mnstate.edu/peil/MDEV102/U1/S6/Complement3.htm|access-date=2020-09-04|website=web.mnstate.edu}}</ref> is the set of [[Element (mathematics)|elements]] not in {{mvar|A}}.<ref name=":1">{{Cite web|title=Complement (set) Definition (Illustrated Mathematics Dictionary)|url=https://www.mathsisfun.com/definitions/complement-set-.html|access-date=2020-09-04|website=www.mathsisfun.com}}</ref>

When all elements in the [[Universe (set theory)|universe]], i.e. all elements under consideration, are considered to be [[Element (mathematics)|members]] of a given set {{mvar|U}}, the '''absolute complement''' of {{mvar|A}} is the set of elements in {{mvar|U}} that are not in {{mvar|A}}.

The '''relative complement''' of {{mvar|A}} with respect to a set {{mvar|B}}, also termed the '''set difference''' of {{mvar|B}} and {{mvar|A}}, written <math>B \setminus A,</math> is the set of elements in {{mvar|B}} that are not in {{mvar|A}}.