Parity Multiplication is Associative/Proof 1
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Theorem
Let $R := \struct {\set {\text{even}, \text{odd} }, +, \times}$ be the parity ring.
The operation $\times$ is associative:
- $\forall a, b, c \in R: \paren {a \times b} \times c = a \times \paren {b \times c}$
Proof
From Isomorphism between Ring of Integers Modulo 2 and Parity Ring:
- $\struct {\set {\text{even}, \text{odd} }, +, \times}$ is isomorphic with $\struct {\Z_2, +_2, \times_2}$
the ring of integers modulo $2$.
The result follows from:
and:
$\blacksquare$