# Definition:Smallest Element

## Definition

Let $\struct {S, \preceq}$ be an ordered set.

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

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

That is, $x$ strictly precedes, or is equal to, every element of $S$.

The Smallest Element is Unique, so calling it the smallest element is justified.

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

For an element to be the smallest element, all $y \in S$ must be comparable with $x$.

### Smallest Element of Subset

Let $\struct {S, \preceq}$ be an ordered set.

Let $T \subseteq S$ be a subset of $S$.

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

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

where $\preceq \restriction_T$ denotes the restriction of $\preceq$ to $T$.

### Class Theory

In the context of class theory, the definition follows the same lines:

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:

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

## Comparison with Minimal Element

Compare the definition of 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 defined as

Some sources do not bother to define this concept on a general ordered set and instead apply it directly to a totally ordered set or even a well-ordered set.

## 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 - see above)
• 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 $\FF \setminus \O$, 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.