Bounds and Interval Notation

The Big Idea

A bound is a value that a set does not pass. A lower bound stays at or below every member of the set; an upper bound stays at or above every member. Interval notation then records the region in which the allowed values lie.

\[ 2<x\le7 \qquad\Longleftrightarrow\qquad x\in(2,7]. \]

Here \(2\) is the greatest lower bound, but it is not a member of the set. The value \(7\) is the least upper bound and is a member. That one difference—whether a limiting value is attained—separates a bound from a minimum or maximum.

What Is a Bound?

Let \(S\) be a set of real numbers. A number \(L\) is a lower bound of \(S\) when every member of \(S\) is at least \(L\):

\[ L\le x\qquad\text{for every }x\in S. \]

A number \(U\) is an upper bound of \(S\) when every member of \(S\) is at most \(U\):

\[ x\le U\qquad\text{for every }x\in S. \]

A bound is a statement about the entire set. Checking only a few convenient members cannot establish a bound unless those members represent all possible cases.

One Set Usually Has Many Bounds

Consider

\[ S=[2,5]. \]

The numbers \(5\), \(6\), \(10\), and \(1000\) are all upper bounds because no member of \(S\) exceeds them. Likewise, \(2\), \(1\), \(0\), and \(-1000\) are all lower bounds.

Bounds do not need to be close to the set. But close bounds usually communicate more information. Saying a machine part weighs at most \(2.1\) kilograms is more useful than saying it weighs at most \(100\) kilograms, even if both statements are true.

Bounded Above, Bounded Below, and Bounded

In some contexts, “unbounded” means unbounded in at least one direction. When the direction matters, say unbounded above, unbounded below, or unbounded in both directions.

A Single Absolute-Value Bound

A real set \(S\) is bounded exactly when all its members fit within some finite distance of zero. In symbols, there is a real number \(M\ge0\) such that

\[ |x|\le M\qquad\text{for every }x\in S. \]

For example, every \(x\in[-3,7]\) satisfies \(|x|\le7\). The one magnitude bound \(M=7\) supplies both a lower and an upper bound:

\[ -7\le x\le7. \]

The enclosing interval need not be the smallest possible one. It only needs to contain the set.

Minimum and Maximum

A minimum is the smallest member of a set. A maximum is the largest member.

\[ m=\min S \quad\Longleftrightarrow\quad m\in S\text{ and }m\le x\text{ for every }x\in S, \]

\[ M=\max S \quad\Longleftrightarrow\quad M\in S\text{ and }x\le M\text{ for every }x\in S. \]

Membership is essential. A value cannot be the minimum or maximum of a set unless it belongs to that set.

A Set May Approach a Bound Without Reaching It

The open interval \((2,5)\) contains numbers as close to \(2\) or \(5\) as we wish, but it contains neither boundary value.

It has no minimum. If someone proposes a member \(x>2\) as the smallest, the number

\[ \frac{x+2}{2} \]

is still greater than \(2\), belongs to \((2,5)\), and is smaller than \(x\). The same reasoning shows that \((2,5)\) has no maximum.

“No minimum” does not mean the set extends downward forever. It means the limiting lower value is not attained.

Infimum and Supremum

The infimum of a nonempty set is its greatest lower bound. The supremum is its least upper bound.

\[ \inf S=\text{greatest lower bound of }S, \]

\[ \sup S=\text{least upper bound of }S. \]

For \(S=(2,5)\),

\[ \inf S=2, \qquad \sup S=5, \]

even though \(2\notin S\) and \(5\notin S\). Infimum and supremum describe the tight limiting bounds; they do not require membership.

How the Four Extremal Ideas Fit Together

Whenever a minimum exists, it equals the infimum. Whenever a maximum exists, it equals the supremum:

\[ \min S=\inf S, \qquad \max S=\sup S. \]

The converses require membership. If \(\inf S\notin S\), there is no minimum; if \(\sup S\notin S\), there is no maximum.

Sharp and Loose Bounds

A sharp or tight bound cannot be improved while remaining valid. The infimum is the sharp lower bound, and the supremum is the sharp upper bound.

For \(S=[-3,4]\), the statement \(x\le10\) gives a valid but loose upper bound. The statement \(x\le4\) gives the sharp upper bound.

In applied work, a loose bound may still be useful if it is easy to prove or sufficient for a decision. Engineering, algorithms, probability, and approximation often balance the cost of finding a tighter estimate against the benefit that tighter information provides.

Interval Notation as a Bound Record

Interval notation packages lower and upper boundary information into a compact form. Let \(a<b\).

For all four intervals, \(\inf S=a\) and \(\sup S=b\). Brackets determine whether those sharp bounds are also the minimum or maximum.

Endpoints and Bounds Are Related but Different

An endpoint is a boundary value used to describe an interval. A bound is any value that limits an entire set from below or above.

One-Sided Bounds and Infinite Intervals

Infinity is not a real bound or an endpoint. It indicates the absence of a finite boundary in that direction, so interval notation always uses a parenthesis beside \(\infty\) or \(-\infty\).

How to Show That a Set Is Unbounded

A set is unbounded above when no proposed real upper bound works. To show this, explain how to find a member larger than any proposed bound.

The positive integers \(S=\{1,2,3,\ldots\}\) are unbounded above. Given any proposed upper bound \(U\), there is an integer \(n>U\). Therefore \(U\) cannot bound the set.

Similarly, the set of all integers \(\mathbb Z\) is unbounded both above and below. This is stronger than merely observing that the written list appears to continue.

Finite Sets Always Attain Their Bounds

Every nonempty finite set of real numbers has a minimum and maximum. For

\[ F=\{-6,-1,3,10\}, \]

we have

\[ \min F=\inf F=-6, \qquad \max F=\sup F=10. \]

Infinite sets behave differently. They may or may not be bounded, and a sharp bound may exist without being attained.

An Infinite Set with a Missing Lower Limit

Let

\[ S=\left\{\frac1n:n\in\mathbb N,\ n\ge1\right\} =\left\{1,\frac12,\frac13,\frac14,\ldots\right\}. \]

The set has maximum \(1\). Its members get arbitrarily close to \(0\) but never equal \(0\), so

\[ \sup S=\max S=1, \qquad \inf S=0, \qquad \min S\text{ does not exist}. \]

This is a useful early model for limits: approaching a value is not the same as reaching it.

Why the Real Numbers Are Complete

The least-upper-bound property says that every nonempty subset of \(\mathbb R\) that is bounded above has a real supremum. Equivalently, every nonempty set bounded below has a real infimum.

This property is called completeness. It rules out gaps in the real number system and supports limits, continuity, area, optimization, and calculus.

The rational numbers do not have this property when considered by themselves. For example,

\[ Q=\{q\in\mathbb Q:q^2<2\} \]

is bounded above in \(\mathbb Q\), but its sharp boundary \(\sqrt2\) is not rational. The real numbers fill that gap.

The Smallest Interval Covering a Bounded Set

If a nonempty set \(S\subseteq\mathbb R\) is bounded, its infimum and supremum determine the smallest closed interval containing it:

\[ S\subseteq[\inf S,\sup S]. \]

This enclosing interval is sometimes called the set’s interval hull. For

\[ S=\{-4,1,3,8\}, \]

the interval hull is \([-4,8]\). The hull fills the gaps between members, so it is generally larger than the original set.

Translate Bounds Among Three Forms

Words such as at least, at most, no less than, and no more than include the stated boundary. Words such as greater than, below, and strictly between exclude it.

Bounds Depend on the Domain

The inequality \(2<x<6\) describes different sets over different domains:

\[ \{x\in\mathbb R:2<x<6\}=(2,6), \]

\[ \{x\in\mathbb Z:2<x<6\}=\{3,4,5\}. \]

Over \(\mathbb R\), the set has no minimum or maximum. Over \(\mathbb Z\), it has minimum \(3\) and maximum \(5\). The same written bounds can have different attained extrema because the available points differ.

Singletons, Empty Sets, and Reversed Bounds

When lower and upper boundaries coincide,

\[ [a,a]=\{a\}, \qquad (a,a)=\varnothing. \]

A condition such as \(8\le x\le2\) is impossible because its proposed lower bound exceeds its proposed upper bound. Its solution set is empty.

The empty set requires special care in advanced work. Bound and extremum conventions for \(\varnothing\) may depend on whether one is working only in \(\mathbb R\) or in the extended real numbers. At this level, treat minimum and maximum as nonexistent and state the ambient convention before assigning an infimum or supremum.

Open, Closed, and Bounded Answer Different Questions

These properties should not be blended together.

“Open” does not mean infinite, and “closed” does not mean finite. Boundedness concerns extent; openness and closedness concern boundary or limit-point behavior in the surrounding space.

Combining Bounds by Addition and Subtraction

Suppose

\[ a\le x\le b, \qquad c\le y\le d. \]

Adding the smallest possible values and the largest possible values gives

\[ a+c\le x+y\le b+d. \]

For subtraction, the smallest result uses the smallest \(x\) and largest \(y\), while the largest uses the largest \(x\) and smallest \(y\):

\[ a-d\le x-y\le b-c. \]

In interval form,

\[ [a,b]+[c,d]=[a+c,b+d]. \]

Scaling and Reversing Bounds

If \(a\le x\le b\) and \(k>0\), multiplication by \(k\) preserves order:

\[ ka\le kx\le kb. \]

If \(k<0\), the inequality directions reverse and the endpoint order must be restored:

\[ kb\le kx\le ka. \]

For example, \(2\le x\le5\) implies

\[ -15\le-3x\le-6. \]

Products and quotients of two variable intervals require sign analysis. Testing only the two matching endpoints can miss the true extrema; all relevant endpoint combinations and zero restrictions must be considered.

Bounding a Function

To bound \(f(x)\) on an input set, determine the smallest and largest possible outputs—not merely the smallest and largest inputs.

On \([-2,3]\), the function \(f(x)=x^2\) has

\[ 0\le x^2\le9. \]

The maximum \(9\) occurs at the endpoint \(x=3\), but the minimum \(0\) occurs at the interior point \(x=0\). Simply squaring the two endpoints would give \(4\) and \(9\) and would miss the true lower bound.

For a monotone increasing function, endpoint values do give the output bounds on a closed interval. For a function that changes direction, inspect critical points, corners, discontinuities, and relevant limits as well.

Guaranteed Bounds and Estimated Ranges

A guaranteed bound follows from definitions, data limits, or proved reasoning. An estimated range is a prediction that may fail.

If a tank’s certified capacity is \(500\) liters, then its contents satisfy \(V\le500\text{ L}\) under normal use. If a manager expects demand between \(420\) and \(480\) liters, \([420,480]\) is a forecast interval, not a physical guarantee.

Labeling the source and confidence of a range prevents a planning estimate from being mistaken for a hard constraint.

Bounds Hidden by Rounding

A displayed measurement usually represents a range of possible underlying values. If a positive quantity rounds to \(12.3\) to the nearest tenth under the usual round-half-up rule, then

\[ 12.25\le x<12.35. \]

Thus the display \(12.3\) represents the half-open interval \([12.25,12.35)\) under that rule. A different tie-breaking rule can change endpoint membership, so the rounding convention matters at exact halfway values.

When the tie rule is unknown or exact halfway cases are not material, the simpler error statement

\[ |x-12.3|\le0.05 \]

provides a safe closed enclosure, although it may include a boundary value that a specific rounding rule assigns elsewhere.

Absolute and Relative Error Bounds

If an approximation \(a\) has absolute error at most \(E\), then the true value \(x\) satisfies

\[ |x-a|\le E \qquad\Longleftrightarrow\qquad x\in[a-E,a+E]. \]

If \(a=80\) and \(E=2\), then \(x\in[78,82]\).

Relative error compares the error with a reference magnitude. For nonzero true value \(x\),

\[ \text{relative error}=\frac{|a-x|}{|x|}. \]

State whether the denominator uses the true value, an accepted value, or an approximation. Different fields may adopt different operational conventions.

Bounds in Practical Constraints

Physical context may supply additional bounds. Weight is nonnegative, exam scores may have a fixed maximum, and a calendar date belongs to a discrete domain even when time itself is modeled continuously.

Bounds Prepare Us for Optimization

Optimization asks for the best attainable value under stated constraints. Bounds tell us what “best” could mean and whether the limiting value is reachable.

• A maximum is an attained best upper value.
• A supremum is the sharp upper limit, whether attained or not.
• A minimum is an attained best lower value.
• An infimum is the sharp lower limit, whether attained or not.

A business may have a supremum for theoretical output that machinery can approach but never reach because downtime is unavoidable. Calling that value a maximum would overstate what can actually occur.

A Reliable Bound-Finding Routine

1. Identify the set and domain. Know exactly which values are allowed.
2. Inspect both directions. Ask whether the set extends indefinitely below or above.
3. Propose valid bounds. Verify that each bound works for every member.
4. Tighten the bounds. Determine whether a closer valid bound exists.
5. Test membership. Decide whether each sharp boundary belongs to the set.
6. Name the result precisely. Distinguish bound, infimum, supremum, minimum, and maximum.
7. Write the interval or set notation. Preserve inclusion, gaps, and the intended domain.
8. Check the context. Confirm units, rounding rules, physical restrictions, and whether the bounds are guaranteed or estimated.

Common Confusions

Working Summary

A lower bound stays at or below an entire set, and an upper bound stays at or above it. A set is bounded when it has finite bounds on both sides. Ordinary bounds need not be tight or belong to the set.

The infimum is the greatest lower bound, and the supremum is the least upper bound. When a sharp bound belongs to the set, it becomes the minimum or maximum. This membership test explains why open intervals can be bounded without having attained extrema.

Interval notation records boundary location and inclusion. Bounds then support inequality reasoning, error control, measurement, range analysis, function behavior, optimization, and later work with limits and completeness.

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