# 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

- 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