When studying sets of numbers, one often needs to single out the largest or the smallest value belonging to a given set.
The notions of maximum and minimum make precisely this intuitive idea precise, and they are among the first fundamental tools one encounters in mathematical analysis.
In the sections that follow we shall introduce the rigorous definitions of the maximum and minimum of a set, study their main properties, and work through several instructive examples.
Contents
- Maximum of a set
- Minimum of a set
- Uniqueness of the maximum and the minimum
- When do the maximum and minimum exist?
- Examples
- Relationship with the supremum and infimum
Maximum of a set
Let \(A\subseteq\mathbb{R}\) be a non-empty set.
An element \(M\in A\) is called the maximum of \(A\) if
\[ x\leq M \qquad \forall x\in A. \]
In other words, the maximum is the largest element of the set, that is, an element that is greater than or equal to every other element of the set.
When it exists, we write
\[ M=\max A. \]
Saying that \(M\) is the maximum of \(A\) therefore amounts to verifying two conditions at once:
- \(M\in A\);
- \(x\leq M\) for every \(x\in A\).
The first condition is essential: a number that does not belong to the set cannot be its maximum.
Minimum of a set
Let \(A\subseteq\mathbb{R}\) be a non-empty set.
An element \(m\in A\) is called the minimum of \(A\) if
\[ m\leq x \qquad \forall x\in A. \]
The minimum is thus the smallest element of the set, that is, an element that is less than or equal to every other element of the set.
When it exists, we write
\[ m=\min A. \]
Here too the following conditions must hold simultaneously:
- \(m\in A\);
- \(m\leq x\) for every \(x\in A\).
Uniqueness of the maximum and the minimum
If a set has a maximum, then it is unique.
Indeed, suppose that \(M_1\) and \(M_2\) are both maxima of the set.
Since \(M_1\) is a maximum,
\[ M_2\leq M_1. \]
Likewise, since \(M_2\) is a maximum,
\[ M_1\leq M_2. \]
From the two inequalities it follows that
\[ M_1=M_2. \]
Hence the two maxima coincide.
The same argument shows that the minimum, when it exists, is likewise unique.
When do the maximum and minimum exist?
Not every set has a maximum or a minimum.
For a set to have a maximum, there must exist an element of the set that is greater than or equal to every other element of the set.
Similarly, for a set to have a minimum, there must exist an element of the set that is less than or equal to every other element of the set.
Whether a maximum or a minimum exists therefore depends not only on the shape of the set, but also on whether the candidate extreme value actually belongs to the set.
Examples
Closed interval
Consider the interval
\[ [1,5]. \]
The left endpoint \(1\) belongs to the interval and is less than or equal to every other element of it.
Hence
\[ \min[1,5]=1. \]
Similarly,
\[ \max[1,5]=5. \]
Open interval
Now consider
\[ (1,5). \]
The numbers \(1\) and \(5\) do not belong to the interval.
Consequently,
\[ \min(1,5) \]
does not exist, and
\[ \max(1,5) \]
does not exist either.
No matter how close one gets to \(5\), it is always possible to find an element of the interval that is larger still.
The same holds near \(1\).
A set with a maximum but no minimum
Consider
\[ A=(0,1]. \]
Since \(1\in A\) and no element of \(A\) is greater than \(1\),
\[ \max A=1. \]
However, \(0\notin A\).
Moreover, there is no element of the set that is less than or equal to every other element of the set.
Hence the minimum does not exist.
A set with a minimum but no maximum
Consider
\[ [2,+\infty). \]
The number \(2\) belongs to the set and is less than or equal to every other element of it.
Therefore,
\[ \min[2,+\infty)=2. \]
The set, on the other hand, has no maximum, since it contains arbitrarily large numbers.
Relationship with the supremum and infimum
The notions of maximum and minimum are closely related to those of supremum and infimum.
In particular:
- if the maximum of a set exists, then it coincides with its supremum;
- if the minimum of a set exists, then it coincides with its infimum.
The converse, however, does not always hold.
For example, the open interval
\[ (1,5) \]
has no maximum, yet it does admit the number \(5\) as its supremum.
Similarly, it has no minimum, yet it admits the number \(1\) as its infimum.
The notions of maximum and minimum are closely linked to those of supremum and infimum. When they exist, the maximum coincides with the supremum, just as the minimum coincides with the infimum. The converse, however, fails: a set may have a supremum without having a maximum (as happens for the open interval \((1,5)\), whose supremum is \(5\)).