Integer Addition is Cancellable
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Theorem
The operation of addition on the set of integers $\Z$ is cancellable:
- $\forall x, y, z \in \Z: x + z = y + z \implies x = y$
Proof
Let $x = \eqclass {a, b} {}$, $y = \eqclass {c, d} {}$ and $z = \eqclass {e, f} {}$ for some $x, y, z\in \Z$.
Then:
| \(\ds x + z\) | \(=\) | \(\ds y + z\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \eqclass {a, b} {} + \eqclass {e, f} {}\) | \(=\) | \(\ds \eqclass {c, d} {} + \eqclass {e, f} {}\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \eqclass {a + e, b + f} {}\) | \(=\) | \(\ds \eqclass {c + e, d + f} {}\) | Definition of Integer Addition | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds a + e\) | \(=\) | \(\ds c + e\) | Definition of Integer | ||||||||||
| \(\, \ds \land \, \) | \(\ds b + f\) | \(=\) | \(\ds d + f\) | Definition of Integer | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds a\) | \(=\) | \(\ds c\) | Natural Number Addition is Cancellable | ||||||||||
| \(\, \ds \land \, \) | \(\ds b\) | \(=\) | \(\ds d\) | Natural Number Addition is Cancellable | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \eqclass {a, b} {}\) | \(=\) | \(\ds \eqclass {c, d} {}\) | Definition of Integer |
$\blacksquare$
Sources
- 1951: Nathan Jacobson: Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts ... (previous) ... (next): Introduction $\S 5$: The system of integers