Definition:Positive/Integer
Definition
Informally, the positive integers are the set:
- $\Z_{\ge 0} = \set {0, 1, 2, 3, \ldots}$
As the set of integers $\Z$ is the Inverse Completion of Natural Numbers, it follows that elements of $\Z$ are the isomorphic images of the elements of equivalence classes of $\N \times \N$ where two tuples are equivalent if the difference between the two elements of each tuple is the same.
Thus positive can be formally defined on $\Z$ as a relation induced on those equivalence classes as specified in the definition of integers.
That is, the integers being defined as all the difference congruence classes, positive can be defined directly as the relation specified as follows:
- The integer $z \in \Z: z = \eqclass {\tuple {a, b} } \boxminus$ is positive if and only if $b \le a$.
The set of positive integers is denoted $\Z_{\ge 0}$.
An element of $\Z$ can be specifically indicated as being positive by prepending a $+$ sign:
- $+x := x \in \Z_{\ge 0}$.
Ordering on Integers
Definition 1
The integers are ordered on the relation $\le$ as follows:
- $\forall x, y \in \Z: x \le y$
- $\exists c \in P: x + c = y$
where $P$ is the set of positive integers.
That is, $x$ is less than or equal to $y$ if and only if $y - x$ is non-negative.
Definition 2
The integers are ordered on the relation $\le$ as follows:
Let $x$ and $y$ be defined as from the formal definition of integers:
- $x = \eqclass {x_1, x_2} {}$ and $y = \eqclass {y_1, y_2} {}$ where $x_1, x_2, y_1, y_2 \in \N$.
Then:
- $x < y \iff x_1 + y_2 \le x_2 + y_1$
where:
- $+$ denotes natural number addition
- $\le$ denotes natural number ordering.
Also known as
As there is often confusion as to whether or not $0$ is included in the set of positive integers, it may be preferable to refer to the set of non-negative integers instead.
The notation $\Z^+$ is common, but leaves it ambiguous as to whether $\Z_{>0}$ or $\Z_{\ge 0}$ is meant.
Also see
Sources
- 1951: Nathan Jacobson: Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts ... (previous) ... (next): Introduction $\S 5$: The system of integers
- 1964: J. Hunter: Number Theory ... (previous) ... (next): Chapter $\text {I}$: Number Systems and Algebraic Structures: $1$. Introduction
- 1971: Wilfred Kaplan and Donald J. Lewis: Calculus and Linear Algebra ... (previous) ... (next): Introduction: Review of Algebra, Geometry, and Trigonometry: $\text{0-1}$: The Real Numbers
- 1979: G.H. Hardy and E.M. Wright: An Introduction to the Theory of Numbers (5th ed.) ... (previous) ... (next): $\text I$: The Series of Primes: $1.1$ Divisibility of integers
- 1988: Dominic Welsh: Codes and Cryptography ... (previous) ... (next): Notation