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 $S$ can be denoted:
- $\map \min S$
- $0$
- $\mathrm O$
or similar.
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 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 defined as
Some sources do not bother to define the concept of a smallest element 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)
- 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
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 14$: Order
- 1964: W.E. Deskins: Abstract Algebra ... (previous) ... (next): $\S 2.3$: Definition $2.3$
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings
- 1968: A.N. Kolmogorov and S.V. Fomin: Introductory Real Analysis ... (previous) ... (next): $\S 3.5$: Well-ordered sets. Ordinal Numbers: Definition $1$
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 7$
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 2$: Continuum Property: $\S 2.7$: Maximum and Minimum
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.5$: Ordered Sets
- 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $2$: Partial Order Relations
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): minimum (plural minima)
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): null element
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): minimum (plural minima)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): null element
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): least