Quotient Structure is Well-Defined
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Theorem
Let $\struct {S, \circ}$ be an algebraic structure.
Let $\RR$ be a congruence relation on $\struct {S, \circ}$.
Let $S / \RR$ be the quotient set of $S$ by $\RR$.
Let $\circ_\RR$ be the operation induced on $S / \RR$ by $\circ$.
Then $\circ_\RR$ is a well-defined operation in the quotient structure $\struct {S / \RR, \circ_\RR}$.
Proof
\(\ds \eqclass {x_1} \RR = \eqclass {x_2} \RR\) | \(\land\) | \(\ds \eqclass {y_1} \RR = \eqclass {y_2} \RR\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds x_1 \mathop \RR x_2\) | \(\land\) | \(\ds y_1 \mathop \RR y_2\) | Definition of Equivalence Class | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {x_1 \circ y_1}\) | \(\RR\) | \(\ds \paren {x_2 \circ y_2}\) | Definition of Congruence Relation | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \eqclass {x_1 \circ y_1} \RR\) | \(=\) | \(\ds \eqclass {x_2 \circ y_2} \RR\) | Definition of Equivalence Class |
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 11$: Quotient Structures
- 1974: Thomas W. Hungerford: Algebra ... (previous) ... (next): $\S 1.1$