Addition of Cross-Relation Equivalence Classes on Natural Numbers is Cancellable

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Theorem

Let $\struct {\N, +}$ be the semigroup of natural numbers under addition.

Let $\struct {\N \times \N, \oplus}$ be the (external) direct product of $\struct {\N, +}$ with itself, where $\oplus$ is the operation on $\N \times \N$ induced by $+$ on $\N$.


Let $\boxtimes$ be the cross-relation defined on $\N \times \N$ by:

$\tuple {x_1, y_1} \boxtimes \tuple {x_2, y_2} \iff x_1 + y_2 = x_2 + y_1$


Let $\eqclass {x, y} {}$ denote the equivalence class of $\tuple {x, y}$ under $\boxtimes$.

The operation $\oplus$ on these equivalence classes is cancellable, in the sense that:

\(\displaystyle \eqclass {a, b} {} \oplus \eqclass {c_1, d_1} {}\) \(=\) \(\displaystyle \eqclass {a, b} {} \oplus \eqclass {c_2, d_2} {}\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \eqclass {c_1, d_1} {}\) \(=\) \(\displaystyle \eqclass {c_2, d_2} {}\)

and similarly:

\(\displaystyle \eqclass {c_1, d_1} {} \oplus \eqclass {a, b} {}\) \(=\) \(\displaystyle \eqclass {c_2, d_2} {} \oplus \eqclass {a, b} {}\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \eqclass {c_1, d_1} {}\) \(=\) \(\displaystyle \eqclass {c_2, d_2} {}\)


Proof

\(\displaystyle \eqclass {a, b} {} \oplus \eqclass {c_1, d_1} {}\) \(=\) \(\displaystyle \eqclass {a, b} {} \oplus \eqclass {c_2, d_2} {}\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \eqclass {b, a} {} \oplus \paren {\eqclass {a, b} {} \oplus \eqclass {c_1, d_1} {} }\) \(=\) \(\displaystyle \eqclass {b, a} {} \oplus \paren {\eqclass {a, b} {} \oplus \eqclass {c_2, d_2} {} }\) adding $\eqclass {b, a} {}$ to both sides
\(\displaystyle \leadsto \ \ \) \(\displaystyle \paren {\eqclass {b, a} {} \oplus \eqclass {a, b} {} } \oplus \eqclass {c_1, d_1} {}\) \(=\) \(\displaystyle \paren {\eqclass {b, a} {} \oplus \eqclass {a, b} {} } \oplus \eqclass {c_2, d_2} {}\) Integer Addition is Associative
\(\displaystyle \leadsto \ \ \) \(\displaystyle \eqclass {a + b, a + b} {} \oplus \eqclass {c_1, d_1} {}\) \(=\) \(\displaystyle \eqclass {a + b, a + b} {} \oplus \eqclass {c_2, d_2} {}\) Definition of $\oplus$
\(\displaystyle \leadsto \ \ \) \(\displaystyle \eqclass {c_1, d_1} {}\) \(=\) \(\displaystyle \eqclass {c_2, d_2} {}\) Identity for Addition of Cross-Relation Equivalence Classes on Natural Numbers

$\blacksquare$

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