Minimum of a Set¶

Definition¶

Let $E\subseteq \mathbb{R}$ be a set. The minimum of $E\text{,}$ denoted by $\min E\text{,}$ is the smallest element in $E$ if it exists. That is, $\min E = x$ if $x\in E$ and $x\leq y$ for all $y\in E\text{.}$

Interpretation¶

The minimum of a set represents the smallest value that is actually attained within the set.

It formalizes the idea of a smallest element, distinguishing it from lower bounds that may not belong to the set.

The set $S=\{1,\tfrac{1}{2},\tfrac{1}{3},\dots\}$ has no minimum. Its elements approach $0$, but $0\notin S$$S$, so $\inf S=0$ is not attained.

The defining inequality compares the proposed minimum with every element, and membership requires that the proposed value actually occur. A lower bound outside the set is therefore not a minimum.

Key Results¶

• A set can have at most one minimum.
• The minimum, if it exists, is always the greatest lower bound for the set.
• A set may have lower bounds without having a minimum.

Examples¶

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

Prerequisites¶

• Inequality

Related Concepts¶

• Maximum of a Set
• Lower Bound
• Infimum

Sources¶

• Math I: Calculus · Lecture 3

Aliases: least element, smallest element, minimum element Symbols: element of, real numbers, subset

Last updated: 17 October 2026 21:53