Inverse for Integer Addition
Theorem
Each element $x$ of the set of integers $\Z$ has an inverse element $-x$ under the operation of integer addition:
- $\forall x \in \Z: \exists -x \in \Z: x + \paren {-x} = 0 = \paren {-x} + x$
Proof
Let us define $\eqclass {\tuple {a, b} } \boxtimes$ as in the formal definition of integers.
That is, $\eqclass {\tuple {a, b} } \boxtimes$ is an equivalence class of ordered pairs of natural numbers under the congruence relation $\boxtimes$.
$\boxtimes$ is the congruence relation defined on $\N \times \N$ by:
- $\tuple {x_1, y_1} \boxtimes \tuple {x_2, y_2} \iff x_1 + y_2 = x_2 + y_1$
In order to streamline the notation, we will use $\eqclass {a, b} {}$ to mean $\eqclass {\tuple {a, b} } \boxtimes$, as suggested.
From the method of construction, the element $\eqclass {a, a + x} {}$ has an inverse $\eqclass {a + x, a} {}$ where $a$ and $x$ are elements of the natural numbers $\N$.
Thus:
\(\ds \eqclass {a, a + x} {} + \eqclass {a + x, a} {}\) | \(=\) | \(\ds \eqclass {a + a + x, a + x + a} {}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \eqclass {a, a} {}\) | Construction of Inverse Completion: Members of Equivalence Classes | |||||||||||
\(\ds \) | \(=\) | \(\ds \eqclass {a + x + a , a + a + x} {}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \eqclass {a + x, a} {} + \eqclass {a, a + x} {}\) |
So $\eqclass {a, a + x} {}$ has the inverse $\eqclass {a + x, a} {}$.
This article, or a section of it, needs explaining. In particular: Needs more detail explaining the connection to the natural numbers. The Inverse Completion page is okay, but the link to what's going on here is hard to follow. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code. |
$\blacksquare$
Sources
- 1951: Nathan Jacobson: Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts ... (previous) ... (next): Introduction $\S 5$: The system of integers
- 1964: W.E. Deskins: Abstract Algebra ... (previous) ... (next): $\S 2.5$: Theorem $2.25$
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $2$: Integers and natural numbers: $\S 2.1$: The integers: $\mathbf Z. \, 4$
- 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): $\S 1$