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Maximum of a Set

Definition

Let E⊆RE\subseteq \mathbb{R} be a set. The maximum of E,E\text{,} denoted by max⁡E,\max E\text{,} is the largest element in EE if it exists. That is, max⁡E=x\max E= x if x∈Ex\in E and x≥yx\geq y for all y∈E.y\in E\text{.}

Interpretation

The maximum of a set is its largest element: a value that belongs to the set and is at least as large as every other element. Not every set has one: a set can be bounded above without any element being on top.

Two number lines: the closed interval from 0 to 1, whose maximum 1 belongs to the set, and the open interval from 0 to 1, which has no largest element although its supremum is 1.

Both sets in the figure lie between 00 and 11. The closed interval [0,1][0,1] contains 11, so its maximum is 11. The open interval (0,1)(0,1) has no maximum: for any x∈(0,1)x\in(0,1), the midpoint 12(x+1)\tfrac{1}{2}(x+1) also lies in (0,1)(0,1) and is larger. Its supremum is still 11, but 1∉(0,1)1\notin(0,1).

The definition has two parts. The inequality x≥yx\ge y for all y∈Ey\in E says that xx is an upper bound; the condition x∈Ex\in E says that this bound is attained. An upper bound outside the set is therefore not a maximum. The two parts also give uniqueness: if xx and x′x' are both maxima, then x≥x′x\ge x' and x′≥xx'\ge x.

A set has a maximum if and only if it has a supremum that belongs to the set, and then max⁡E=sup⁡E\max E=\sup E. Every nonempty finite set has a maximum; infinite sets such as (0,1)(0,1) or N\mathbb{N} may have none. The minimum of a set is the mirror image. Later, the extreme value theorem gives conditions under which the set of values of a function has a maximum.

A common slip is to write max⁡E\max E where only sup⁡E\sup E exists. The set {1−1n ⁣:n≥1}\{1-\tfrac{1}{n}\colon n\ge1\} has supremum 11 but no maximum, since every element is below 11.

Key Results

  • A set can have at most one maximum.
  • The maximum, if it exists, is the least upper bound for the set.
  • A set may have upper bounds without having a maximum.

Examples

  • The set {1, 2, 3}\{1,\,2,\,3\} has maximum 3.3\text{.}
  • The interval (0,1)(0,1) has no maximum.

Prerequisites

Sources

Aliases: greatest element, largest element, maximum element Symbols: element of, natural numbers, real numbers, subset

Last updated: 20 October 2026 03:02