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Version 2 |
| 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 {\em sum of functions} $f$ and $g$ as the following function \;$fg\!: A \to M$: |
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 {\em sum of functions} $f$ and $g$ as the following function \;$fg\!: A \to M$: |
| $$(f\!+\!g)(x) = f(x)\!+\!g(x) \quad \forall x \in A$$ |
$$(f\!+\!g)(x) = f(x)\!+\!g(x) \quad \forall x \in A$$ |
| If $r$ is any element of the ring $R$, then the {\em scalar-multiplied function}\; $rf\!: A \to M$\, is defined as |
If $r$ is any element of the ring $R$, then the {\em scalar-multiplied function}\; $rf\!: A \to M$\, is defined as |
| $$(rf)(x) = r\!\cdot\!f(x) \quad \forall x \in A.$$ |
$$(rf)(x) = r\!\cdot\!f(x) \quad \forall x \in A.$$ |
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| 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 {\em product of functions} $f$ and $g$ as the function\; $fg\!: A \to K$ as follows: |
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 {\em product of functions} $f$ and $g$ as the function\; $fg\!: A \to K$ as follows: |
| $$(fg)(x) = f(x)\!\cdot\!g(x) \quad \forall x \in A$$ |
$$(fg)(x) = f(x)\!\cdot\!g(x) \quad \forall x \in A$$ |
| The {\em quotient of functions} $f$ and $g$ is the function\; |
The {\em 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 |
$\displaystyle\frac{f}{g}\!: \{a\in A\,\vdots\;\; g(a) \neq 0\} \to K$\; defined as |
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$$\frac{f}{g}(x) = \frac{f(x)}{g(x)} \quad \forall x \in A\!\smallsetminus\!\{a\in A\,\vdots\;\; g(a) = 0\}.$$
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$$\frac{f}{g}(x) = \frac{f(x)}{g(x)} \quad \forall x \in A\!\smallsetminus\!\{a\in A\,\vdots\;\; g(a) \neq 0\}.$$
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| For example, we can with a \PMlinkescapetext{clear} conscience say that the \PMlinkname{tangent}{TrigonometricFunction} function is the quotient of the \PMlinkname{sine}{TrigonometricFunction} and the cosine functions. |
For example, we can with a \PMlinkescapetext{clear} conscience say that the \PMlinkname{tangent}{TrigonometricFunction} function is the quotient of the \PMlinkname{sine}{TrigonometricFunction} and the cosine functions. |
