Associative Operation/Examples/Associative/x circ a circ y
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Example of Associative Operation
Let $\struct {S, \circ}$ be an algebraic structure where $\circ$ is an associative operation.
Let $a \in S$ be an arbitrary element of $S$.
Let $*$ be the operation defined on $S$ by:
- $\forall x, y \in S: x * y := x \circ a \circ y$
Then $*$ is associative on $S$.
Proof
Let $x, y, z \in S$ be arbitrary.
\(\ds x * \paren {y * z}\) | \(=\) | \(\ds x \circ a \circ \paren {y \circ a \circ z}\) | Definition of $*$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {x \circ a \circ y} \circ a \circ z\) | as $\circ$ is associative | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {x * y} * z\) | Definition of $*$ |
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 2$: Compositions: Exercise $2.15$