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 $13.12 \ \text{(d)}$