Complement (set theory)

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.

If

A

is the area colored red in this image…

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.

… then the complement of

A

is everything else.

In set theory, the complement of a set $A$ , often denoted by $A^{\complement }$ (or $A'$ ),^[1] is the set of elements not in $A$ .^[2]

When all elements in the universe, i.e. all elements under consideration, are considered to be members of a given set $U$ , the absolute complement of $A$ is the set of elements in $U$ that are not in $A$ .

The relative complement of $A$ with respect to a set $B$ , also termed the set difference of $B$ and $A$ , written $B\setminus A,$ is the set of elements in $B$ that are not in $A$ .

^ "Complement and Set Difference". web.mnstate.edu. Retrieved 2020-09-04.
^ "Complement (set) Definition (Illustrated Mathematics Dictionary)". www.mathsisfun.com. Retrieved 2020-09-04.

[1] "Complement and Set Difference". web.mnstate.edu. Retrieved 2020-09-04.

[:1-2] "Complement (set) Definition (Illustrated Mathematics Dictionary)". www.mathsisfun.com. Retrieved 2020-09-04.

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