# 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:

\(\displaystyle x + z\) | \(=\) | \(\displaystyle y + z\) | |||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \eqclass {a, b} {} + \eqclass {e, f} {}\) | \(=\) | \(\displaystyle \eqclass {c, d} {} + \eqclass {e, f} {}\) | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \eqclass {a + e, b + f} {}\) | \(=\) | \(\displaystyle \eqclass {c + e, d + f} {}\) | Definition of Integer Addition | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle a + e\) | \(=\) | \(\displaystyle c + e\) | Definition of Integer | |||||||||

\(\, \displaystyle \land \, \) | \(\displaystyle b + f\) | \(=\) | \(\displaystyle d + f\) | Definition of Integer | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle a\) | \(=\) | \(\displaystyle c\) | Natural Number Addition is Cancellable | |||||||||

\(\, \displaystyle \land \, \) | \(\displaystyle b\) | \(=\) | \(\displaystyle d\) | Natural Number Addition is Cancellable | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \eqclass {a, b} {}\) | \(=\) | \(\displaystyle \eqclass {c, d} {}\) | 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