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[parent] sum and product and quotient of functions (Definition)

Let $ A$ be a set and $ M$ a left $ R$-module. If $ f\!: A \to M$ and $ g\!: A \to M$, then one may define the sum of functions $ f$ and $ g$ as the following function $ f\!+\!g\!: A \to M$:

$\displaystyle (f\!+\!g)(x) = f(x)\!+\!g(x) \quad \forall x \in A$
If $ r$ is any element of the ring $ R$, then the scalar-multiplied function $ rf\!: A \to M$ is defined as
$\displaystyle (rf)(x) = r\!\cdot\!f(x) \quad \forall x \in A.$

Let $ A$ again be a set and $ K$ a field or a skew field. If $ f\!: A \to K$ and $ g\!: A \to K$, then one can define the product of functions $ f$ and $ g$ as the function $ fg\!: A \to K$ as follows:

$\displaystyle (fg)(x) = f(x)\!\cdot\!g(x) \quad \forall x \in A$
The quotient of functions $ f$ and $ g$ is the function $ \displaystyle\frac{f}{g}\!: \{a\in A\,\vdots\;\; g(a) \neq 0\} \to K$ defined as
$\displaystyle \frac{f}{g}(x) = \frac{f(x)}{g(x)} \quad \forall x \in A\!\smallsetminus\!\{a\in A\,\vdots\;\; g(a) = 0\}.$

In particular, the incremental quotient of functions $ \frac{f(y)-f(x)}{y-x}$, as $ y$ tends to $ x$, gave rise to the important concept of derivative. As another example, we can with a clear conscience say that the tangent function is the quotient of the sine and the cosine functions.



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See Also: direct sum of even/odd functions (example), limit rules of functions, polynomial function, proof of limit rule of product, continuous derivative implies bounded variation

Also defines:  sum of functions, product of functions, quotient of functions, scalar-multiplied function

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Cross-references: cosine, quotient, derivative, skew field, field, ring, function
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This is version 8 of sum and product and quotient of functions, born on 2008-01-13, modified 2008-02-29.
Object id is 10190, canonical name is ProductAndQuotientOfFunctionsSum.
Accessed 1636 times total.

Classification:
AMS MSC: 03E20 (Mathematical logic and foundations :: Set theory :: Other classical set theory )
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