Linear Combination Operator on Real Numbers is Entropic

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Theorem

Let $a, b \in \R$ be real numbers.

Let $\odot$ be the operation on $\R$ defined as:

$\forall x, y \in \R: x \odot y := a x + b y$

Then $\odot$ is an entropic operation.


Proof

\(\ds \forall p, q, r, s \in \R: \, \) \(\ds \paren {p \odot q} \odot \paren {r \odot s}\) \(=\) \(\ds a \paren {a p + b q} + b \paren {a r + b s}\) Definition of $\odot$
\(\ds \) \(=\) \(\ds a^2 p + a b q + b a r + b^2 s\) Real Multiplication Distributes over Addition
\(\ds \) \(=\) \(\ds a^2 p + a b r + b a q + b^2 s\) Real Multiplication is Commutative and Real Addition is Commutative
\(\ds \) \(=\) \(\ds a \paren {a p + b r} + b \paren {a q + b s}\) Real Multiplication Distributes over Addition
\(\ds \) \(=\) \(\ds \paren {p \odot r} \odot \paren {q \odot s}\) Real Multiplication Distributes over Addition

$\blacksquare$

Sources

  • 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 13$: Compositions Induced on Cartesian Products and Function Spaces: Exercise $13.12 \ \text{(d)}$