Addition of Cross-Relation Equivalence Classes on Natural Numbers is Well-Defined

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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 well-defined, in the sense that:

\(\displaystyle \eqclass {a_1, b_1} {}\) \(=\) \(\displaystyle \eqclass {a_2, b_2} {}\)
\(\displaystyle \eqclass {c_1, d_1} {}\) \(=\) \(\displaystyle \eqclass {c_2, d_2} {}\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \eqclass {a_1, b_1} {} \oplus \eqclass {c_1, d_1} {}\) \(=\) \(\displaystyle \eqclass {a_2, b_2} {} \oplus \eqclass {c_2, d_2} {}\)


Proof

Let $\eqclass {a_1, b_1} {}, \eqclass {a_2, b_2} {}, \eqclass {c_1, d_1} {}, \eqclass {c_2, d_2} {}$ be $\boxtimes$-equivalence classes such that $\eqclass {a_1, b_1} {} = \eqclass {a_2, b_2} {}$ and $\eqclass {c_1, d_1} {} = \eqclass {c_2, d_2} {}$.


Then:

\(\displaystyle \eqclass {a_1, b_1} {}\) \(=\) \(\displaystyle \eqclass {a_2, b_2} {}\) Definition of Operation Induced by Direct Product
\(\, \displaystyle \land \, \) \(\displaystyle \eqclass {c_1, d_1} {}\) \(=\) \(\displaystyle \eqclass {c_2, d_2} {}\) Definition of Operation Induced by Direct Product
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle a_1 + b_2\) \(=\) \(\displaystyle a_2 + b_1\) Definition of Cross-Relation
\(\, \displaystyle \land \, \) \(\displaystyle c_1 + d_2\) \(=\) \(\displaystyle c_2 + d_1\) Definition of Cross-Relation


Then we have:

\(\displaystyle \tuple {a_1 + c_1} + \tuple {b_2 + d_2}\) \(=\) \(\displaystyle \tuple {a_1 + b_2} + \tuple {c_1 + d_2}\) Commutativity and associativity of $+$
\(\displaystyle \) \(=\) \(\displaystyle \tuple {a_2 + b_1} + \tuple {c_2 + d_1}\) from above: $a_1 + b_2 = a_2 + b_1, c_1 + d_2 = c_2 + d_1$
\(\displaystyle \) \(=\) \(\displaystyle \tuple {a_2 + c_2} + \tuple {b_1 + d_1}\) Commutativity and associativity of $+$
\(\displaystyle \leadsto \ \ \) \(\displaystyle \tuple {a_1 + c_1, b_1 + d_1}\) \(\boxtimes\) \(\displaystyle \tuple {a_2 + c_2, b_2 + d_2}\) Definition of $\boxtimes$
\(\displaystyle \leadsto \ \ \) \(\displaystyle \tuple {\tuple {a_1, b_1} \oplus \tuple {c_1, d_1} }\) \(\boxtimes\) \(\displaystyle \tuple {\tuple {a_2, b_2} \oplus \tuple {c_2, d_2} }\) Definition of $\oplus$

$\blacksquare$

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