Left Operation is Right Distributive over All Operations
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Theorem
Let $\struct {S, \circ, \leftarrow}$ be an algebraic structure where:
- $\leftarrow$ is the left operation
- $\circ$ is any arbitrary binary operation.
Then $\leftarrow$ is right distributive over $\circ$.
Proof
\(\ds \forall a, b, c \in S: \paren {a \circ b} \leftarrow c\) | \(=\) | \(\ds a \circ b\) | Definition of Left Operation | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {a \leftarrow c} \circ \paren {b \leftarrow c}\) | Definition of Left Operation |
The result follows by definition of right distributivity.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 16$: The Natural Numbers: Exercise $16.23 \ \text{(a)}$