Intervals on the Real Line

What Makes a Set an Interval?

In the previous tutorial, absolute-value inequalities produced solution sets lying between two endpoints or outside them. Intervals give a precise way to describe many of these sets. The central idea is simple: if a set contains two real numbers, it contains every real number between them.

This idea depends on the order of the real line, not on a formula or a particular pair of endpoints. A set can be an interval even when its endpoints are not included, and a set with a gap is not an interval. We will define this between-points property, learn the basic notation, and prove how intervals behave under intersections and unions.

The definition allows a singleton, such as \(\{2\}\), to be an interval: there are no two distinct elements \(x,z\) in it, so the between-points condition holds. The empty set also satisfies the condition vacuously. In contrast, a set with two elements but without some number between them fails the condition.

The definition is about what happens between points of the set. It does not require that the set have a smallest or largest element. For example, the positive real numbers contain every real number strictly between any two of their elements, even though the set has no smallest positive real number. Endpoint notation makes these distinctions visible.

Notation for Common Intervals

For real numbers \(a<b\), the standard notation for bounded intervals is:

A parenthesis marks an endpoint that is excluded, and a square bracket marks one that is included. These notations are consistent with the inequalities in the set descriptions: for example, \(a\leq x\) includes \(a\), so the left endpoint uses a square bracket.

Intervals can also extend without bound. For example, \((a,\infty)\) means \(\{x\in\mathbb{R}:x>a\}\), and \((-\infty,b]\) means \(\{x\in\mathbb{R}:x\leq b\}\). The symbols \(\infty\) and \(-\infty\) describe unboundedness; they are not real endpoints, so they always appear with parentheses. The whole real line \(\mathbb{R}\) is an interval, as is any ray of one of these forms.

These are familiar examples, not a definition that limits intervals to the displayed forms. The between-points property is the general test. The next tutorial examines open and closed intervals in more detail; here, the emphasis is on recognizing intervals and using their basic structural properties.

The Interval Between Two Points

A basic way to use the definition is to look at all real numbers between two specified points. If \(a\leq b\), the closed segment \([a,b]\) contains both points and every point between them. It is also the smallest interval that contains both: any interval containing \(a\) and \(b\) must contain every intermediate point.

Proof. First, let \(u,v\in[a,b]\) and suppose \(u<y<v\). Since \(a\leq u\) and \(v\leq b\), transitivity gives \(a<y<b\), and hence \(y\in[a,b]\). Thus \([a,b]\) is an interval. This reasoning also covers \(a=b\): in that case there cannot be \(u,v\in[a,b]\) with \(u<y<v\), so the interval condition holds.

Now suppose \(I\) is an interval containing \(a\) and \(b\). If \(a=b\), then \([a,b]=\{a\}\subseteq I\). If \(a<b\), the endpoints \(a,b\) belong to \(I\), and every \(y\) with \(a<y<b\) belongs to \(I\) by the definition of interval. Thus every element of \([a,b]\) belongs to \(I\), proving \([a,b]\subseteq I\). \(\square\)

This theorem explains why the closed segment is a useful reference point even when a particular interval excludes one or both endpoints. Once two points are known to lie in an interval, everything strictly between them must lie there too.

Intersections and Unions of Intervals

Intersections preserve the between-points property. If two points lie in every set in a collection of intervals, then every point between them lies in every one of those intervals as well. This remains true for any collection, not just a finite one.

Proof. Let \(\mathcal{I}\) be a collection of intervals, and let \(J=\bigcap_{I\in\mathcal{I}} I\). If \(J\) is empty, it is an interval by convention. Otherwise, take \(x,z\in J\) and \(y\in\mathbb{R}\) with \(x<y<z\). Because \(x,z\in J\), both \(x\) and \(z\) belong to every \(I\in\mathcal{I}\). Each \(I\) is an interval, so \(y\in I\) for every \(I\in\mathcal{I}\). Therefore \(y\in J\). This proves that \(J\) is an interval. \(\square\)

Unions require more care. Two intervals separated by a gap can have a union that is not an interval, as the previous example showed. But if two intervals share at least one point, their union is an interval: the shared point connects the two sets, leaving no gap between points in different intervals.

Proof. Choose \(c\in I\cap J\). Take \(x,z\in I\cup J\) with \(x<y<z\). If \(x,z\in I\), then \(y\in I\) because \(I\) is an interval. If \(x,z\in J\), then \(y\in J\). It remains to consider the case where the endpoints are not both in the same interval. If necessary, interchange the names of \(I\) and \(J\); we can then assume \(x\in I\) and \(z\in J\).

If \(y\leq c\), then \(x<y\leq c\). Both \(x\) and \(c\) belong to \(I\), so the interval property gives \(y\in I\) (and if \(y=c\), this also follows directly from \(c\in I\)). If \(c<y\), then \(c<y<z\), and \(c,z\in J\), so \(y\in J\). In either case \(y\in I\cup J\). Thus \(I\cup J\) is an interval. \(\square\)

Inequality Solutions as Intervals

The interval viewpoint is especially useful when expressing solution sets. A two-sided inequality often describes one interval; an exterior inequality often describes two separate rays, whose union has a gap. The endpoint symbols record whether equality is allowed in the original inequality.

An exterior condition illustrates the contrasting structure. For instance, \(|x-1|>2\) is equivalent to \(x<-1\) or \(x>3\). These two rays do not form a single interval: \(1\) belongs to neither ray but lies between \(-2\) and \(4\), which do belong to their union. The between-points test detects the gap immediately.

A Practical Test and a Common Pitfall

To decide whether a proposed set is an interval, do not rely only on how its formula looks. Instead, ask whether there could be two members of the set with a missing real number between them. If so, the set is not an interval. If the set is described by bounds, check that every number between any two allowed values still meets the bounds.

A common error is to assume that every union of intervals is an interval. The union theorem proved here needs the intervals to intersect. For example, \([0,1]\cup[3,4]\) has a gap, while \([0,3]\cup[2,4]\) has a nonempty intersection and is an interval. More generally, the fact that each piece is an interval does not by itself rule out missing numbers between the pieces.

The between-points definition is also distinct from endpoint notation. Notation such as \((a,b]\) describes a particular interval and says exactly which endpoints it contains. The definition applies to arbitrary subsets, including sets described in other ways, and lets us prove that they are intervals without first putting them into a named form.