Definition:Smallest Element/Class Theory

Definition

Let $V$ be a basic universe.

Let $\RR \subseteq V \times V$ be an ordering.

Let $A$ be a subclass of the field of $\RR$.

An element $x \in A$ is the smallest element of $A$ if and only if:

$\forall y \in A: x \mathrel \RR y$

The smallest element of $A$ is denoted $\min A$.

Comparison with Minimal Element

Compare the definition of a minimal element with that of a smallest element.

Consider the ordered set $\struct {S, \preceq}$ such that $T \subseteq S$.

An element $x \in T$ is the smallest element of $T$ if and only if:

$\forall y \in T: x \preceq y$

That is, $x$ is comparable with, and precedes, or is equal to, every $y \in T$.

An element $x \in T$ is a minimal element of $T$ if and only if:

$y \preceq x \implies x = y$

That is, $x$ precedes, or is equal to, every $y \in T$ which is comparable with $x$.

If all elements are comparable with $x$, then such a minimal element is indeed the smallest element.

Note that when an ordered set is in fact a totally ordered set, the terms minimal element and smallest element are equivalent.

Also known as

The smallest element of a collection is also called:

• The least element
• The lowest element (particularly with numbers)
• The first element
• The minimum element (but beware confusing with minimal)
• The null element (in the context of boolean algebras and boolean rings)

Examples

Finite Subsets of Natural Numbers

Let $\FF$ denote the set of finite subsets of the natural numbers $\N$.

Consider the ordered set $\struct {\FF, \subseteq}$.

Then $\struct {\FF, \subseteq}$ has a smallest element, and that is the empty set $\O$.

Finite Subsets of Natural Numbers less Empty Set

Let $\FF$ denote the set of finite subsets of the natural numbers $\N$.

Let $\GG$ denote the set $\FF \setminus \set \O$, that is, $\FF$ with the empty set excluded.

Consider the ordered set $\struct {\GG, \subseteq}$.

$\struct {\GG, \subseteq}$ has no smallest element.

Also see

• Definition:Greatest Element (Class Theory)

• Results about smallest elements can be found here.

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text I$ -- Superinduction and Well Ordering: $\S 1$ Introduction to well ordering