Right Operation is Associative

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

• Left Operation is Associative

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 $2.9$