Left Operation is Associative
Theorem
The left operation is associative:
- $\forall x, y, z: \paren {x \gets y} \gets z = x \gets \paren {y \gets z}$
Proof
| \(\ds x \gets \paren {y \gets z}\) | \(=\) | \(\ds x \gets y\) | Definition of Left Operation | |||||||||||
| \(\ds \) | \(=\) | \(\ds x\) | Definition of Left Operation |
| \(\ds \paren {x \gets y} \gets z\) | \(=\) | \(\ds x \gets z\) | Definition of Left Operation | |||||||||||
| \(\ds \) | \(=\) | \(\ds x\) | Definition of Left Operation |
$\blacksquare$
Also see
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 2$: Compositions: Example $2.4$
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 2$: Compositions: Exercise $2.9$