Definition:Smallest/Ordered Set

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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 $T$.


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 set 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.


Sources