Definition:Positive

Definition
Let $\left({R, +, \circ, \le}\right)$ be an ordered ring whose zero is $0_R$.

Then $x \in R$ is positive iff $0_R \le x$.

The set of all positive elements of $R$ is denoted:


 * $R_+ := \left\{{x \in R: 0_R \le x}\right\}$

Integers
Informally, the positive integers are the set:
 * $Z_+ = \left\{{0, 1, 2, 3, \ldots}\right\}$

As the set of integers 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 Unique Minus 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 Unique Minus congruence classes, positive can be defined directly as the relation specified as follows:


 * The integer $z \in \Z: z = \left[\!\left[{\left({a, b}\right)}\right]\!\right]_\boxminus$ is positive iff $b \le a$.

The set of positive integers is denoted $\Z_+$.

An element of $\Z$ can be specifically indicated as being positive by prepending a $+$ sign:


 * $+x \in \Z \iff x \in \Z_+$.

Ordering on Integers
The integers are ordered on the relation $<$ as follows:


 * $\forall x, y \in \Z: y - x \in \Z_+ \iff x \le y$

That is, $x$ is less than or equal to $y$ iff $y - x$ is positive.