T1: Infimum and Minimum

December 31, 2024

• Definition 1.
• Definition 2.
• Difference
• Reference

We see that infimum and minimum are basic the same thing. Most of people may not be clear about the difference between them. This post introduces the difference and help researcher have a good writing.

Definition 1.

A lower bound of a subset $S$ of a partially ordered set $P$ is an element $x\in P$ such that $x \leq z$ for all $z\in S$.

Definition 2.

We say that $a$ is is an infimum of $S$ if for all lower bounds $z\in P$ of $S$, $x\leq a$. (greatest lower bound)

Difference

Example:

• For $S = \{3, 4, 5\}$, the infimum is 3 (same as the minimum).
• For $S=(3,4)$(open interval), the infimum is 3, but it is not part of the set. There is no minimum.

Reference

[1]. Wiki. Infimum and supremum