Supremum vs Maximum

Excerpt

Give several illustrative examples to didactically clarify the difference between a supremum and a maximum as explained in this short primer:

---

Give several illustrative examples to didactically clarify the difference between a supremum and a maximum as explained in this short primer:

---

A maximum of a set must be an element of the set. A supremum need not be.

Explicitly, if $X$ is a (partially) ordered set, and $S$ is a subset, then an element $s\_0$ is the supremum of $S$ if and only if:

1. $$\begin{gathered} s \\ leqs\_0 \end{gathered}$$ for all $$\begin{gathered} s \\ inS \end{gathered}$$; and
2. If $$\begin{gathered} t \\ inX \end{gathered}$$ is such that $$\begin{gathered} s \\ leqt \end{gathered}$$ for all $$\begin{gathered} s \\ inS \end{gathered}$$, then $$\begin{gathered} s\_0 \\ leqt \end{gathered}$$.

By contrast, an element $m$ is the maximum of $S$ if and only if:

1. $$\begin{gathered} s \\ leqm \end{gathered}$$ for all $$\begin{gathered} s \\ inS \end{gathered}$$; and
2. $$\begin{gathered} m \\ inS \end{gathered}$$.

Note that if $S$ has a maximum, then the maximum must be the supremum: indeed, if $$\begin{gathered} t \\ inX \end{gathered}$$ is such that $$\begin{gathered} s \\ leqt \end{gathered}$$ for all $$\begin{gathered} s \\ inS \end{gathered}$$, then in particular $$\begin{gathered} m \\ inS \end{gathered}$$, so $$\begin{gathered} m \\ leqt \end{gathered}$$, proving that $m$ satisfies the conditions to be the supremum.

But it is possible for a set to have a supremum but not a maximum. For instance, in the real numbers, the set of all negative numbers does not have a maximum: there is no negative number $m$ with the property that $$\begin{gathered} n \\ leqm \end{gathered}$$ for all negative numbers $n$. However, the set of all negative numbers does have a supremum: $0$ is the supremum of the set of negative numbers. Indeed, $$\begin{gathered} a \\ leq0 \end{gathered}$$ for all negative numbers $a$; and if $$\begin{gathered} a \\ leqb \end{gathered}$$ for all negative numbers $a$, then $$\begin{gathered} 0 \\ leqb \end{gathered}$$.

The full relationship between supremum and maximum is:

1. If $S$ has a maximum $m$, then $S$ also has a supremum and in fact $m$ is also a supremum of $S$.
2. Conversely, if $S$ has a supremum $s$, then $S$ has a maximum if and only if $$\begin{gathered} s \\ inS \end{gathered}$$, in which case the maximum is also $s$.

In particular, if a set has both a supremum and a maximum, then they are the same element. The set may also have neither a supremum nor a maximum (e.g., the rationals as a subset of the reals). But if it has only one them, then it has a supremum which is not a maximum and is not in the set.

Response from GPT-4o

To clarify the difference between a supremum and a maximum, let’s consider several illustrative examples:

Example 1: The Set of Negative Real Numbers

Set: ( S = { x \in \mathbb{R} \mid x < 0 } )

• Supremum: The supremum of ( S ) is 0. This is because 0 is greater than or equal to all elements of ( S ), and no number less than 0 can serve as an upper bound for ( S ).
• Maximum: ( S ) does not have a maximum because there is no single largest negative number. For any negative number you choose, there is always another negative number that is larger but still negative.

Example 2: The Set of Rational Numbers Less Than √2

Set: ( S = { x \in \mathbb{Q} \mid x < \sqrt{2} } )

• Supremum: The supremum of ( S ) is ( \sqrt{2} ). This is because ( \sqrt{2} ) is greater than or equal to all elements of ( S ), and no rational number can be an upper bound that is smaller than ( \sqrt{2} ).
• Maximum: ( S ) does not have a maximum because ( \sqrt{2} ) is irrational, and thus not in the set of rational numbers. There is no largest rational number less than ( \sqrt{2} ).

Example 3: The Set of Real Numbers Less Than or Equal to 1

Set: ( S = { x \in \mathbb{R} \mid x \leq 1 } )

• Supremum: The supremum of ( S ) is 1. This is because 1 is greater than or equal to all elements of ( S ), and there is no number smaller than 1 that serves as an upper bound for ( S ).
• Maximum: The maximum of ( S ) is also 1 because 1 is an element of ( S ) and it is the largest such element.

Example 4: The Set of Natural Numbers

Set: ( S = { x \in \mathbb{N} } = { 0, 1, 2, 3, \ldots } )

• Supremum: The supremum of ( S ) is not finite. Since the set of natural numbers is unbounded, we consider the supremum to be infinity (∞) in the extended real number system.
• Maximum: ( S ) does not have a maximum because natural numbers are infinite and there is no largest natural number.

Example 5: The Set of Real Numbers Between 0 and 1 (Inclusive)

Set: ( S = { x \in \mathbb{R} \mid 0 \leq x \leq 1 } )

• Supremum: The supremum of ( S ) is 1. This is because 1 is greater than or equal to all elements of ( S ), and there is no number smaller than 1 that serves as an upper bound for ( S ).
• Maximum: The maximum of ( S ) is also 1 because 1 is an element of ( S ) and it is the largest such element.

Example 6: The Set of Real Numbers Between 0 and 1 (Exclusive)

Set: ( S = { x \in \mathbb{R} \mid 0 < x < 1 } )

• Supremum: The supremum of ( S ) is 1. This is because 1 is greater than or equal to all elements of ( S ), and there is no number smaller than 1 that serves as an upper bound for ( S ).
• Maximum: ( S ) does not have a maximum because 1 is not an element of ( S ) and there is no largest number in the interval ( (0, 1) ).

Summary

• A maximum of a set must be an element of the set and is the largest element within the set.
• A supremum (least upper bound) of a set is the smallest number that is greater than or equal to every element in the set. It does not need to be an element of the set.
• If a set has a maximum, that maximum is also the supremum.
• If a set has a supremum but no maximum, the supremum is not an element of the set.