Sets - Complement of a Set

Definition of Complement: The complement of a set $A$, denoted by $A'$, is the set of all elements of the Universal set $U$ which are not elements of $A$. Mathematically, $A' = \{x : x \in U \text{ and } x \notin A\}$. Visually, if the Universal set is a rectangle and set $A$ is a circle inside it, $A'$ is represented by the entire shaded region inside the rectangle but outside the circle.

Complement Relative to Universal Set: The complement of a set depends entirely on the Universal set $U$ provided. For instance, if $A$ is the set of even numbers, its complement $A'$ would be the set of odd numbers if $U$ is the set of natural numbers, but it could be different if $U$ is the set of integers.

Complement Laws: The union of a set $A$ and its complement $A'$ always results in the Universal set $U$ ($A \cup A' = U$). Conversely, the intersection of a set and its complement is always an empty set ($A \cap A' = \emptyset$), meaning they are disjoint. Visually, this shows that the circle $A$ and its exterior $A'$ share no space and together fill the bounding rectangle.

Law of Double Complementation: If you take the complement of a complement, you return to the original set, expressed as $(A')' = A$. In a Venn diagram, this is equivalent to 'un-shading' the exterior and 're-shading' the interior circle.

Laws of Empty Set and Universal Set: The complement of the Universal set is the empty set ($U' = \emptyset$), and the complement of the empty set is the Universal set ($\emptyset' = U$). This indicates that nothing exists outside the boundary of $U$, and everything in $U$ exists outside the boundary of nothingness.

De Morgan's First Law: The complement of the union of two sets is equal to the intersection of their complements, written as $(A \cup B)' = A' \cap B'$. Visually, if you take two overlapping circles and look at the area outside their combined shape, it is identical to finding the area that is simultaneously outside circle $A$ and outside circle $B$.

De Morgan's Second Law: The complement of the intersection of two sets is equal to the union of their complements, written as $(A \cap B)' = A' \cup B'$. Visually, the region representing everything except the small overlapping 'almond' shape between two circles is the same as combining all area outside $A$ with all area outside $B$.