Right Operation is Associative
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Theorem
The right operation is associative:
- $\forall x, y, z: \paren {x \to y} \to z = x \to \paren {y \to z}$
Proof
\(\ds \paren {x \to y} \to z\) | \(=\) | \(\ds y \to z\) | Definition of Right Operation | |||||||||||
\(\ds \) | \(=\) | \(\ds z\) | Definition of Right Operation |
\(\ds x \to \paren {y \to z}\) | \(=\) | \(\ds x \to z\) | Definition of Right Operation | |||||||||||
\(\ds \) | \(=\) | \(\ds z\) | Definition of Right 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 $2.9$