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Section0.3Other important sets

We have seen a few important sets above — namely \(\mathbb{N}, \mathbb{Z}, \mathbb{Q}\) and \(\mathbb{R}\text{.}\) However, arguably the most important set in mathematics is the empty set.

Definition0.3.1Empty set

The empty set (or null set or void set) is the set which contains no elements. It is denoted \(\es\text{.}\) For any object \(x\text{,}\) we always have \(x \notin \es\text{;}\) hence \(\es = \{ \}\text{.}\)

Note that it is important to realise that the empty set is not nothing; think of it as an empty bag. Also note that with quite a bit of hard work you can actually define the natural numbers in terms of the empty set. Doing so is very formal and well beyond the scope of this text.

When a set does not contain too many elements it is fine to specify it by listing out its elements. But for infinite sets or even just big sets we can't do this and instead we have to give the defining rule. For example the set of all perfect square numbers we write as

\begin{align*} S &= \{x \st x = k^2 \mbox{ where } k \in \mathbb{Z} \} \end{align*}

Notice we have used another piece of short-hand here, namely \(\st\text{,}\) which stands for “such that” or “so that”. We read the above statement as “\(S\) is the set of elements \(x\) such that \(x\) equals \(k\)-squared where \(k\) is an integer”. This is the standard way of writing a set defined by a rule, though there are several shorthands for “such that”. We shall use two them:

\begin{align*} P &= \{ p \st p \mbox{ is prime} \} = \{ p \,|\, p \mbox{ is prime} \} \end{align*}

Other people also use “:” is shorthand for “such that”. You should recognise all three of these shorthands.

In this text we will use “\(|\)” or “\(\st\)” (being the preference of the authors), but some other texts use “:”. You should recognise all of these.

The sets \(A\) and \(B\) in the above example illustrate an important point. Every element in \(B\) is an element in \(A\text{,}\) and so we say that \(B\) is a subset of \(A\)

Definition0.3.3

Let \(A\) and \(B\) be sets. We say “\(A\) is a subset of \(B\)” if every element of \(A\) is also an element of \(B\text{.}\) We denote this \(A \subseteq B\) (or \(B \supseteq A\)). If \(A\) is a subset of \(B\) and \(A\) and \(B\) are not the same , so that there is some element of \(B\) that is not in \(A\) then we say that \(A\) is a proper subset of \(B\text{.}\) We denote this by \(A \subset B\) (or \(B \supset A\)).

Two things to note about subsets:

  • Let \(A\) be a set. It is always the case that \(\es \subseteq A\text{.}\)
  • If \(A\) is not a subset of \(B\) then we write \(A \not\subseteq B\text{.}\) This is the same as saying that there is some element of \(A\) that is not in \(B\text{.}\) That is, there is some \(a \in A\) such that \(a \notin B\text{.}\)

In much of our work with functions later in the text we will need to work with subsets of real numbers, particularly segments of the “real line”. A convenient and standard way of representing such subsets is with interval notation.

Definition0.3.5Open and closed intervals of \(\mathbb{R}\)

Let \(a, b \in \mathbb{R}\) such that \(a \lt b\text{.}\) We name the subset of all numbers between \(a\) and \(b\) in different ways depending on whether or not the ends of the interval (\(a\) and \(b\)) are elements of the subset.

  • The closed interval \([a,b] = \{x \in \mathbb{R} : a \leq x \leq b\}\) — both end points are included.
  • The open interval \((a,b) = \{x \in \mathbb{R} : a \lt x \lt b\}\) — neither end point is included.

We also define half-open  2  intervals which contain one end point but not the other:

\begin{align*} (a,b] &= \{ x \in \mathbb{R} : a \lt x \leq b\} & [a,b) &= \{ x \in \mathbb{R} : a \leq x \lt b\} \end{align*}

We sometimes also need unbounded intervals

\begin{align*} [a, \infty) &= \{ x \in \mathbb{R} : a \leq x \} & (a, \infty) &= \{ x \in \mathbb{R} : a \lt x \} \\ (\infty, b] &= \{ x \in \mathbb{R} : x \leq b \} & (\infty, b) &= \{ x \in \mathbb{R} : x \lt b \} \end{align*}

These unbounded intervals do not include “\(\infty\)”, so that end of the interval is always open  3 .