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1) Suppose f is a mathematical quantity that can take on the "states", or "values", f1, f2,...fi,...fn., where n can be finite or infinite.

So, F is the Set { f1, f2,... fi,.. fn } or

F = { f1, f2, fi,... fn }

= { all possible f's}

Now, suppose A is a operator such that when it operates on fi, it gives fj, where i is not equal to j, i.e.

Afi = fj, i =/= j

Afj = fk, j =/= k, and so on

Then can one say anything about A operating on f ?

Af = ?

Note that A is therefore a Mapping that maps F into itself. It maps one element, fi, into a different element fj .

Would it be correct to say that

Af = Mf, where M is some form of modifying factor, that is NOT a function of f ?

2) If Af = Mf is indeed true, then is it necessarily true that A has an Inverse, B, such that

BAf = f ?

If not, then under what conditions will A have such an Inverse?