Non-Zero Integers Closed under Multiplication

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Theorem

The set of non-zero integers is closed under multiplication.


Proof

Let us define $\eqclass {\tuple {a, b} } \boxminus$ as in the formal definition of integers.

That is, $\eqclass {\tuple {a, b} } \boxminus$ is an equivalence class of ordered pairs of natural numbers under the congruence relation $\boxminus$.

$\boxminus$ is the congruence relation defined on $\N \times \N$ as:

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


In order to streamline the notation, we will use $\eqclass {a, b} {}$ to mean $\eqclass {\tuple {a, b} } \boxminus$, as suggested.


Integer multiplication is defined as:

$\forall a, b, c, d \in \N: \eqclass {a, b} {} \times \eqclass {c, d} {} = \eqclass {a c + b d, a d + b c} {}$

From Integer Multiplication is Closed, we have that $x, y \in \Z \implies x y \in \Z$.


From Ring of Integers has no Zero Divisors, we have that $x, y \in \Z: x, y \ne 0 \implies x y \ne 0$.

Therefore multiplication on the non-zero integers is closed.

$\blacksquare$


Sources