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

• Right 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 $2.9$