Greatest Lower Bound of a Set

The greatest lower bound of a non-empty set A is defined as the highest real number that does not exceed any element within A.

This concept is also known as the infimum (inf).

The number, which might or might not belong to set A, functions as a boundary that confines A from below and is symbolized by inf(A).

$$ \inf(A) $$

Defining Greatest Lower Bound

In formal terms, for any non-empty set A comprising real numbers, a number x qualifies as the infimum of A if it fulfills two specific conditions:

1. x acts as a lower bound for A, which means it is less than or equal to every element in A $$ \forall \ a \in A \Rightarrow a \geq x $$
2. x is the highest of all lower bounds for A, implying x exceeds or is equal to any other lower bound y within A $$ \forall \ y \in R \ , \ (\forall \ a \ \in A, a \geq y) \Rightarrow \inf⁡(A) \geq y $$

When the greatest lower bound of A also counts as a member of the set, it earns the title of the minimum of A, denoted as min(A). $$ \inf(A) \in A \Rightarrow \inf(A) = \min(A) $$

Illustrative Examples

Example 1

In the case of the finite set A, where each element is at least 1, 1 serves as a lower bound of A.

$$ A = \{2, 3, 4, 5\} $$

Nonetheless, 2 stands out as the greatest lower bound, thus it's the lower bound of A.

$$ \inf(A) = 2 $$

Here, the lower bound coincides with the minimum of set A.

$$ \inf(A) = \min(A) = 2 $$

Example 2

The set of negative integers, being boundlessly extendable below,

$$ Z^- = \{\ldots, -5, -4, -3, -2, -1 \} $$

has its greatest lower bound at negative infinity (-∞) since a real number lower than any negative integer does not exist.

$$ \inf(Z^-) = -\infty $$

In this scenario, the greatest lower bound is not a part of the set Z^-.

Thus, the set Z^- possesses a greatest lower bound (inf) but lacks a minimum (min).