742: Motion Is Injective

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description/proof of that motion is injective

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Topics

About: vectors space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Natural Language Description
• 3: Proof

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Starting Context

• The reader knows a definition of motion.
• The reader knows a definition of injection.

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Target Context

• The reader will have a description and a proof of the proposition that any motion is injective.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$F_1$: $\in \{\mathbb{R},\mathbb{C}\}$, with the canonical field structure
$F_2$: $\in \{\mathbb{R},\mathbb{C}\}$, with the canonical field structure
$V_1$: $\in \{\text{ the normed }{F}_1\text{ vectors spaces }\}$
$V_2$: $\in \{\text{ the normed }{F}_2\text{ vectors spaces }\}$
$f$: $:V_1\to V_2$, $\in \{\text{ the motions }\}$
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Statements:
$f\in \{\text{ the injections }\}$
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2: Natural Language Description

For any $F_1\in \{\mathbb{R},\mathbb{C}\}$, with the canonical field structure, any $F_2\in \{\mathbb{R},\mathbb{C}\}$, with the canonical field structure, any normed $F_1$ vectors space, $V_1$, any normed $F_2$ vectors space, $V_2$, and any motion, $f:V_1\to V_2$, $f$ is an injection.

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3: Proof

Whole Strategy: Step 1: choose any distinct elements, $v,v^{\prime }\in V_1$; Step 2: suppose that $f(v)=f(v^{\prime })$ and find a contradiction.

Step 1:

Let us choose any elements, $v,v^{\prime }\in V_1$, such that $v\ne v^{\prime }$.

Step 2:

Let us suppose that $f(v)=f(v^{\prime })$.

$\Vert v-v^{\prime }\Vert =\Vert f(v)-f(v^{\prime })\Vert =\Vert 0\Vert =0$, which implies that $v-v^{\prime }=0$, which implies that $v=v^{\prime }$, a contradiction.

So, $f(v)\ne f(v^{\prime })$.

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References

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