The Integers and the Rational Numbers, with the Natural Numbers and the Integers Identified with Subsets of the Rationals

¶ This setting extends Commutative Rings, Fields and Ordered Fields: Standard Notation: its conventions and notation are in force, as extended by the clause identification below.

¶ $\mathbb{Z}$, its operations and order, and its elements $0$ and $1$ (written $0_{\mathbb{Z}}$ and $1_{\mathbb{Z}}$ there) are as in The Integers §integers, The Integers §operations and The Integers §constants. $\mathbb{Z}$ is an ordered ring by The Integers Form an Ordered Ring Containing the Natural Numbers as Its Positive Elements §ordered-ring, so the notation of Commutative Rings, Fields and Ordered Fields: Standard Notation §rings and Commutative Rings, Fields and Ordered Fields: Standard Notation §ordered-rings applies to it.

¶ $\mathbb{Q}$, its operations and order, and its elements $0$ and $1$ (written $0_{\mathbb{Q}}$ and $1_{\mathbb{Q}}$ there) are as in The Rational Numbers §rationals, The Rational Numbers §operations and The Rational Numbers §constants. $\mathbb{Q}$ is an ordered field by The Rational Numbers Form an Archimedean Ordered Field Containing the Integers §ordered-field, so the notation of Commutative Rings, Fields and Ordered Fields: Standard Notation applies to it. The negations of The Integers §operations and The Rational Numbers §operations are those of Negatives, Differences, Reciprocals and Quotients §negative, by The Natural Numbers and the Integers inside the Rational Numbers §negation.

¶ As for $\mathbb{N}_{0}$ in The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion §numbers-only, an element of $\mathbb{Z}$ or of $\mathbb{Q}$ is used as a number only, never as a set.

¶ Let $\iota_{0}:\mathbb{N}_{0}\to\mathbb{Z}$ be as in The Natural Numbers with Zero and Their Embedding into the Integers §embedding and $j:\mathbb{Z}\to\mathbb{Q}$ as in The Rational Numbers §embedding. Both are injective and preserve $0$, $1$, sums, products and the order in both directions, by The Natural Numbers with Zero and Their Embedding into the Integers §embedding, The Integers Form an Ordered Ring Containing the Natural Numbers as Its Positive Elements §embedding and The Rational Numbers Form an Archimedean Ordered Field Containing the Integers §embedding, and so is the composite $j\circ\iota_{0}$; $j$ also preserves negatives, by The Rational Numbers Form an Archimedean Ordered Field Containing the Integers §embedding. The image of $\mathbb{N}_{0}$ under $\iota_{0}$ is the set of nonnegative integers, by The Natural Numbers with Zero and Their Embedding into the Integers §embedding, and the image of $\mathbb{N}$ under $\iota_{0}$ is the set of positive integers, by The Integers Form an Ordered Ring Containing the Natural Numbers as Its Positive Elements §positive. Moreover $\iota_{0}$, $j$ and $j\circ\iota_{0}$ preserve finite sums and products and powers, by Iterated Operations over Finite Sets: Singletons, Disjoint Unions, Reindexing, Products of Sets, Termwise Combination, Homomorphisms and Intervals §homomorphism and Iterated Operations over Finite Sets: Singletons, Disjoint Unions, Reindexing, Products of Sets, Termwise Combination, Homomorphisms and Intervals §reindexing.

¶ The element of $\mathbb{Z}$ or $\mathbb{Q}$ that an $n\in\mathbb{N}_{0}$ denotes by Commutative Rings, Fields and Ordered Fields: Standard Notation §numerals is $\iota_{0}(n)$, respectively $j(\iota_{0}(n))$, by Natural Numbers Read in the Integers and in the Rationals Agree with Their Embeddings §integers and Natural Numbers Read in the Integers and in the Rationals Agree with Their Embeddings §rationals.

¶ An element of $\mathbb{N}_{0}$ or of $\mathbb{Z}$, and a set introduced as a subset of $\mathbb{N}_{0}$ or of $\mathbb{Z}$, is identified with its image under $\iota_{0}$, $j$ or $j\circ\iota_{0}$, as follows. Wherever an element or a subset of $\mathbb{Z}$ or of $\mathbb{Q}$ is required, it stands for its image there; this is the case for an operand next to an operand from $\mathbb{Z}$ or $\mathbb{Q}$, for an operand of a symbol defined on $\mathbb{Z}$ or $\mathbb{Q}$ but not on the smaller set (such as $-x$ and $x-y$ for elements of $\mathbb{N}_{0}$, or $x/y$ and $|x|$ for integers), for a component of a pair or tuple required in a product of copies of $\mathbb{Z}$ or $\mathbb{Q}$, and for an element or a subset of a subset of $\mathbb{Z}$ or $\mathbb{Q}$. The image is taken in the smaller of $\mathbb{Z}$ and $\mathbb{Q}$ that is required; by the clause agreement the choice does not matter. Conversely, wherever an element or a subset of $\mathbb{N}_{0}$, $\mathbb{N}$ or $\mathbb{Z}$ is required, such as an exponent, an index or a limit of a sum, a nonnegative integer stands for the element of $\mathbb{N}_{0}$, a positive integer for the natural number, and a rational number $j(x)$ for the integer $x$, of which it is the image, and a subset contained in such an image stands for the subset of which it is the image, namely its preimage; these readings compose, so that $j(\iota_{0}(n))$ stands for $n$. In this sense $\mathbb{N}\subseteq\mathbb{N}_{0}\subseteq\mathbb{Z}\subseteq\mathbb{Q}$.

¶ By the clause embeddings and by Images of Unions, Intersections, Differences and Subclasses under a Function, and under an Injective Function §union, Images of Unions, Intersections, Differences and Subclasses under a Function, and under an Injective Function §composition, Images of Unions, Intersections, Differences and Subclasses under a Function, and under an Injective Function §membership, Images of Unions, Intersections, Differences and Subclasses under a Function, and under an Injective Function §intersection, Images of Unions, Intersections, Differences and Subclasses under a Function, and under an Injective Function §difference and Images of Unions, Intersections, Differences and Subclasses under a Function, and under an Injective Function §inclusion, this is well defined, and equality, $0$, $1$, sums, products, finite sums and products, powers, negatives and differences (the difference in $\mathbb{N}_{0}$ included, by The Integers Form an Ordered Ring Containing the Natural Numbers as Its Positive Elements §ring), the order, membership in subsets, unions, intersections, set differences and inclusions agree whether they are formed in the smaller set or in the larger one. A subset read in a larger set is a subset of that set, which is then the set $X$ of Sets and Maps: Ordinary Notation §orders: its bounds, suprema and infima are formed there.