Equivalence Relation on Integers Modulo 5 induced by Squaring/Multiplication Modulo Beta is Well-Defined
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Theorem
Let $\beta$ denote the relation defined on the integers $\Z$ by:
- $\forall x, y \in \Z: x \mathrel \beta y \iff x^2 \equiv y^2 \pmod 5$
We have that $\beta$ is an equivalence relation.
Let the $\times_\beta$ operator ("multiplication") on the $\beta$-equivalence classes be defined as:
- $\eqclass a \beta \times_\beta \eqclass b \beta := \eqclass {a \times b} \beta$
Then such an operation is well-defined.
Proof
That $\beta$ is an equivalence relation is proved in Equivalence Relation on Integers Modulo 5 induced by Squaring.
Let:
\(\ds x, x'\) | \(\in\) | \(\ds \eqclass x \beta\) | ||||||||||||
\(\ds y, y'\) | \(\in\) | \(\ds \eqclass y \beta\) |
We have:
\(\ds x^2\) | \(\equiv\) | \(\ds \paren {x'}^2\) | \(\ds \pmod 5\) | |||||||||||
\(\ds y^2\) | \(\equiv\) | \(\ds \paren {y'}^2\) | \(\ds \pmod 5\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {x y}^2\) | \(\equiv\) | \(\ds \paren {x' y'}^2\) | \(\ds \pmod 5\) |
Hence the result.
$\blacksquare$
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $3$: Equivalence Relations and Equivalence Classes: Exercise $9 \ \text{(ii)}$