Theory like coordinate geometry

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One-to-one correspondence is not strictly necessary for theories like coordinate geometry, polar coordinates, and vector analysis to function effectively. Different notations can represent the same object, provided there are clear rules for identifying equivalence between them. For example, the representations of rational numbers and angles can vary while still denoting the same values. In vector analysis, a vector can be represented by a directed line, and its properties can be derived from this representation. Ultimately, the focus is on the relationships and properties shared among the entities being represented.
sadhu
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is one -one correspondence must for a theory like coordinate geometry , polar coordinate ,vector analysis etc
to work , i.e theories which work by representing a quantity by a different set of quantities
behaving alike
 
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It's hard to tell exactly what you're asking... but I think I can answer.

It is fairly common to allow many different notations for the same object; you just have to include rules for identifying when two notations denote the same object.

e.g. 1/2 and 2/4 are two different notations for the same rational number, and 0° and 360° are two different notations for the same angle. (0° and 360° are different angular displacements, of course)
 
what i meant was like in vector we represent a vector by a straight directed line and then use its property to find the property of the represented vector .

this is because they belong to same class of entities i.e vectors

similiarly ordered triplets and points are two quantities which represent each other in space
 

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