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Does anyone know the name given to the number of outputs of a non-injective function that happen to have many (i.e., more than 1) corresponding input (please) ?

Example :

Take $f:\{0,...,9\}\mapsto \{0,...,4\}$ with $f(n)=n \div 2$. Clearly, for each possible output $k$, there is exactly 2 possible inputs $n$ for which $f(n)=k$. There are 5 such situations (for, $k=0,1,2,3,4$)

So here, the ****** of $f$ would be 5, the cardinality of the output set.

What should I write instead of ***** ?

I actually target more complicated functions from one (infinite) vector space to another... is there a name for some kind of ratio between the number outputs corresponding to a single input and the number of those corresponding to multiple inputs ?

thanks !

J

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  • $\begingroup$ I don't think that concept has a name. The phrase "$n$ to $1$" is sometimes used to describe the function. (For interesting functions on vector spaces the size of each image is likely to be $1$ or infinite.) $\endgroup$ – Ethan Bolker Sep 11 '18 at 14:08
  • $\begingroup$ "Cardinality of the image/range" would fit into the *****s. $\endgroup$ – Theo Bendit Sep 11 '18 at 14:26
  • $\begingroup$ @EthanBolker : Ok thanks. Well, I was actually seeking a way to express that some projections from $\mathbb{R}^D$ to $\mathbb{R}^n$ with $n<D$ will be much "harder" to inverse than others, linear projectors being somewhere "in the middle". Something like a "level of injectivity". Functions with high level of injectivity (many outputs having multiple inputs) will be badly reconstructed from outputs, while low injectivity level will "ease" the process. $\endgroup$ – Jerome F Sep 11 '18 at 14:32
  • $\begingroup$ @TheoBendit : can't I create functions with the same (say very high) image cardinality and very different non-injectivity behavior ? What if, on 20 possible inputs, I map 10 inputs to unique outputs and 10 inputs to the same output (giving 11 as image cardinality), or if I evenly map those 10 inputs to those 11 outputs so that no output has a unique associated input ? I would like to measure this difference... $\endgroup$ – Jerome F Sep 11 '18 at 14:40
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    $\begingroup$ In a sense this is the cardinality of the non-invertible elements of the image of $f$ $\endgroup$ – Henry Jun 22 at 10:27

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