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Section 1.8 Complements

Definition 1.8.1.

Let \(S\) be a set, and let \(T \subseteq S\) be a subset. Then the complement of \(T\) in \(S\) is the subset

\begin{equation*} S \setminus T = \{ x \in S : x \notin T \} \subseteq S\text{.} \end{equation*}

Assume that the subset \(T \subseteq S\) is identified using set-builder notation:

\begin{equation*} T = \{x \in S : \phi(x) \}\text{.} \end{equation*}

Then we can do the same for its complement by using negation:

\begin{equation*} S \setminus T = \{x \in S : \neg\phi(x)\}\text{.} \end{equation*}

Consider the set of even natural numbers:

\begin{equation*} T = \{n \in \NN_0 : 2\text{ divides } n\} \subseteq \NN_0\text{.} \end{equation*}

Its complement in \(\NN_0\) is the set of odd integers:

\begin{equation*} \NN_0 \setminus T = \{n \in \NN_0 : 2\text{ does not divide } n\} \subseteq \NN_0\text{.} \end{equation*}

Consider the subset

\begin{equation*} S = \{ x \in \NN_0 : x \geq 173 \}\text{,} \end{equation*}

the set of natural numbers no less than \(173\text{.}\) Its complement in \(\NN_0\) is

\begin{equation*} \NN_0 \setminus S = \{ x \in \NN_0 : x \lt 173 \}\text{,} \end{equation*}

the set of natural numbers strictly less than \(173\text{.}\)

If \(S\) is a set and \(T \subseteq S\) is a subset, you will sometimes see \(S\) referred to as the ambient set. We always form complements in an ambient set, and that ambient set matters. For example, let

\begin{equation*} S_1 = \NN_0 \text{ and } S_2 = \{n \in \NN_0 : n \geq 10\}\text{.} \end{equation*}

Clearly \(S_2 \subset S_1\text{.}\) Now let

\begin{equation*} T = \{n \in \NN_0 : n \geq 11\}\text{.} \end{equation*}

We see that \(T \subset S_2 \subset S_1\text{.}\) Now the complement of \(T\) in \(S_2\) is different from the complement in \(S_1\text{:}\)

\begin{equation*} S_1 \setminus T = \{0,1,2,3,4,5,6,7,8,9,10\} \text{ whereas } S_2 \setminus T = \{10\}\text{.} \end{equation*}