Set Complements

Why

It is often the case in considering set differences that all sets considered are subsets of one set.

Definition

Let $A$ and $B$ denote sets. In many cases, we take the difference between a set and one contained in it. In other words, we assume that $B \subset A$. In this case, we often take complements relative to the same set $A$. So we do not refer to it, and instead refer to the relative complement of $B$ in $A$ as the complement of $B$.

Notation

Let $A$ denote a set, and let $B$ denote a set for which $B \subset A$. We denote the relative complement of $B$ in $A$ by $\relcomplement{B}{A}$. When we need not mention the set $A$, and instead speak of the complement of $B$ without qualification, we denote this complement by $\complement{B}$.

Complement of a complement

One nice property of a complement when $B \subset A$ is:

$(B \subset A) \iff (\relcomplement{\relcomplement{B}{A}}{A} = B)$

Basic facts

Let $E$ denote a set and let $A$ and $B$ denote sets satisfying $A,B \subset E$. Then take all complements with respect to $E$. Here are some immediate consequences of the definition.¹

$\complement{\complement{A}} = A$

$\complement{\emptyset} = E$

$\complement{E} = \varnothing$

$A \subset B \iff \complement{B} \subset \complement{A}$

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