Self-Distributive Structure/Examples/Arithmetic Mean
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Example of Self-Distributive Structure
Let $\Q$ denote the set of rational numbers.
Let $\circ$ be the operation defined on $\Q$ as:
- $\forall x, y \in \Q: x \circ y := \dfrac {x + y} 2$
That is, $x \circ y$ is the arithmetic mean of $x$ and $y$ in $\Q$.
Then the algebraic structure $\struct {\Q, \circ}$ so formed is a self-distributive quasigroup.
Proof
\(\ds \forall a, b, c \in \Q: \, \) | \(\ds a \circ \paren {b \circ c}\) | \(=\) | \(\ds \dfrac {a + \frac {b + c} 2} 2\) | Definition of $\circ$ | ||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac a 2 + \dfrac b 4 + \dfrac c 4\) | simplification | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\dfrac a 4 + \dfrac b 4} + \paren {\dfrac a 4 + \dfrac c 4}\) | rearrangement | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\paren {a \circ b} + \paren {a \circ c} } 2\) | Definition of $\circ$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {a \circ b} \circ \paren {a \circ c}\) | Definition of $\circ$ |
As Rational Addition is Commutative, it follows immediately from Left Distributive and Commutative implies Distributive that:
- $\forall a, b, c \in \Q: \paren {a \circ b} \circ c = \paren {a \circ c} \circ \paren {b \circ c}$
To demonstrate that $\struct {\Q, \circ}$ is a quasigroup, it remains to be shown that:
- $\forall a, b \in \Q: \exists ! x \in \Q: x \circ a = b$
- $\forall a, b \in \Q: \exists ! y \in \Q: a \circ y = b$
We have:
\(\ds x \circ a\) | \(=\) | \(\ds b\) | ||||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \dfrac {x + a} 2\) | \(=\) | \(\ds b\) | |||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds x\) | \(=\) | \(\ds 2 b - a\) |
and similarly:
\(\ds a \circ y\) | \(=\) | \(\ds b\) | ||||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \dfrac {a + y} 2\) | \(=\) | \(\ds b\) | |||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds y\) | \(=\) | \(\ds 2 b - a\) |
Hence both $x$ and $y$ are determined uniquely by $a$ and $b$.
Hence by definition $\struct {\Q, \circ}$ is a quasigroup.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 16$: The Natural Numbers: Exercise $16.21 \ \text{(a)}$