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

## Theorem

Let $\left({\N, +}\right)$ be the semigroup of natural numbers under addition.

Let $\left({\N \times \N, \oplus}\right)$ be the (external) direct product of $\left({\N, +}\right)$ 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:

$\left({x_1, y_1}\right) \boxtimes \left({x_2, y_2}\right) \iff x_1 + y_2 = x_2 + y_1$

Let $\left[\!\left[{x, y}\right]\!\right]$ denote the equivalence class of $\left({x, y}\right)$ under $\boxtimes$.

Let $\otimes$ be the binary operation defined on these equivalence classes as:

$\forall \left[\!\left[{a, b}\right]\!\right], \left[\!\left[{c, d}\right]\!\right] \in \N \times \N: \left[\!\left[{a, b}\right]\!\right] \otimes \left[\!\left[{c, d}\right]\!\right] = \left[\!\left[{\left({a \cdot c}\right) + \left({b \cdot d}\right), \left({a \cdot d}\right) + \left({b \cdot c}\right)}\right]\!\right]$

where $a \cdot c$ denotes natural number multiplication between $a$ and $c$.

The operation $\otimes$ on these equivalence classes is well-defined, in the sense that:

 $\displaystyle \left[\!\left[{a_1, b_1}\right]\!\right]$ $=$ $\displaystyle \left[\!\left[{a_2, b_2}\right]\!\right]$ $\displaystyle \left[\!\left[{c_1, d_1}\right]\!\right]$ $=$ $\displaystyle \left[\!\left[{c_2, d_2}\right]\!\right]$ $\displaystyle \implies \ \$ $\displaystyle \left[\!\left[{a_1, b_1}\right]\!\right] \otimes \left[\!\left[{c_1, d_1}\right]\!\right]$ $=$ $\displaystyle \left[\!\left[{a_2, b_2}\right]\!\right] \otimes \left[\!\left[{c_2, d_2}\right]\!\right]$

## Proof

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

Then:

 $\displaystyle \left[\!\left[{a_1, b_1}\right]\!\right]$ $=$ $\displaystyle \left[\!\left[{a_2, b_2}\right]\!\right]$ Definition of Operation Induced by Direct Product $\, \displaystyle \land \,$ $\displaystyle \left[\!\left[{c_1, d_1}\right]\!\right]$ $=$ $\displaystyle \left[\!\left[{c_2, d_2}\right]\!\right]$ Definition of Operation Induced by Direct Product $\displaystyle \iff \ \$ $\displaystyle a_1 + b_2$ $=$ $\displaystyle a_2 + b_1$ Definition of Cross-Relation $\, \displaystyle \land \,$ $\displaystyle c_1 + d_2$ $=$ $\displaystyle c_2 + d_1$ Definition of Cross-Relation

Both $+$ and $\times$ are commutative and associative on $\N$. Thus:

 $\displaystyle \left({a_1 \cdot c_1 + b_1 \cdot d_1 + a_2 \cdot d_2 + b_2 \cdot c_2}\right)$ $+$ $\displaystyle \left({a_2 \cdot c_1 + b_2 \cdot c_1 + a_2 \cdot d_1 + b_2 \cdot d_1}\right)$ $\, \displaystyle = \,$ $\displaystyle \left({a_1 + b_2}\right) \cdot c_1 + \left({b_1 + a_2}\right) \cdot d_1$ $+$ $\displaystyle a_2 \cdot \left({c_1 + d_2}\right) + b_2 \cdot \left({d_1 + c_2}\right)$ $\, \displaystyle = \,$ $\displaystyle \left({b_1 + a_2}\right) \cdot c_1 + \left({a_1 + b_2}\right) \cdot d_1$ $+$ $\displaystyle a_2 \cdot \left({d_1 + c_2}\right) + b_2 \cdot \left({c_1 + d_2}\right)$ as $a_1 + b_2 = b_1 + a_2, c_1 + d_2 = d_1 + c_2$ $\, \displaystyle = \,$ $\displaystyle b_1 \cdot c_1 + a_2 \cdot c_1 + a_1 \cdot b_2 + a_1 \cdot d_1$ $+$ $\displaystyle a_2 \cdot d_1 + a_2 \cdot c_2 + b_2 \cdot c_1 + b_2 \cdot d_2$ $\, \displaystyle = \,$ $\displaystyle \left({a_1 \cdot d_1 + b_1 \cdot c_1 + a_2 \cdot c_2 + b_2 \cdot d_2}\right)$ $+$ $\displaystyle \left({a_2 \cdot c_1 + b_2 \cdot c_1 + a_2 \cdot d_1 + b_2 \cdot d_1}\right)$

So we have $a_1 \cdot c_1 + b_1 \cdot d_1 + a_2 \cdot d_2 + b_2 \cdot c_2 = a_1 \cdot d_1 + b_1 \cdot c_1 + a_2 \cdot c_2 + b_2 \cdot d_2$ and so, by the definition of $\boxtimes$, we have:

$\left[\!\left[{a_1 \cdot c_1 + b_1 \cdot d_1, a_1 \cdot d_1 + b_1 \cdot c_1}\right]\!\right] = \left[\!\left[{a_2 \cdot c_2 + b_2 \cdot d_2, a_2 \cdot d_2 + b_2 \cdot c_2}\right]\!\right]$

So, by the definition of integer multiplication, this leads to:

$\left[\!\left[{a_1, b_1}\right]\!\right] \otimes \left[\!\left[{c_1, d_1}\right]\!\right] = \left[\!\left[{a_2, b_2}\right]\!\right] \otimes \left[\!\left[{c_2, d_2}\right]\!\right]$

Thus integer multiplication has been shown to be well-defined.

$\blacksquare$