The Relative Complement and Complement of a Set

# The Relative Complement and Complement of a Set

## The Relative Complement of a Set

 Definition: If $A$ is a set, then the Relative Complement of $A$ with respect to $B$ denoted $B \setminus A$ is the set of elements contained $B$ but that are not contained in $A$, that is, $B \setminus A = \{ x : x \in B \: \mathrm{and} \: x \not \in A \}$.

For example, consider the sets $A = \{1, 3, 5, 7, 9 \}$ and $B = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \}$. Then:

(1)
\begin{align} \quad B \setminus A = \{ 2, 4, 6, 8, 10 \} \end{align}

We can represent the relative complement of two sets graphically using Venn diagrams as illustrated below:

We will now state some relatively simple results regarding the relative complement of two sets that the reader is invited to prove if they've never seen the results before:

 Theorem 1: Let $A$ and $B$ be sets. Then: a) $(A \setminus B) \cup B = A \cup B$. b) $(A \setminus B) \cap B = \emptyset$.

## The Complement of a Set

 Definition: If $A$ is a set, then the Complement of $A$ denoted $A^c$ is the set of elements that are not contained in $A$, that is, $A^c = \{ x : x \not \in A \}$.

In general, if $A$ is a set without any context then the complement of $A$ may not be well defined. For example, if we consider the set of real numbers $\mathbb{R}$ then what exactly is $\mathbb{R}^c$? By definition, it is the set of all elements that are not real numbers. Hence, we could say that all cats or all potatoes are contained in $\mathbb{R}^c$, but of course, this sort of inclusion probably isn't the most useful in mathematics.

That said, it must be important to have some sort of context in mind when talking about the complement of a set. If we consider the set of real numbers as a Universal Set, then it is clear from context that the complement of $\mathbb{R}^+$ of nonnegative real numbers is equal to the set of negative real numbers. Hence if a universal set $U$ is understood with respect to $A$, then:

(2)
\begin{align} \quad A^c = U \setminus A \end{align}

We can represent the complement of a set $A$ with respect to some universal set $U$ with the following Venn diagram: