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Hello,
let's consider the set [itex]\Omega[/itex] of all the continuous and integrable functions [itex]f:R \to R[/itex].
Suppose we now take two subsets A and B, where:
- A is the subset of all the gaussian functions centered at the origin: [itex]\exp(-ax^2) [/itex], where a>0
- B is the set of all the even functions: [itex]f(x)=f(-x)[/itex]
It is trivial to prove that [itex]A\subset B[/itex].
However, my question is: is it possible to "quantify" how bigger the subset B is compared to A ?
How should I count the cardinality of A and B ?
My goal is computing the ratio of cardinalities: |B|/|A|. Is it possible?
Thanks.
let's consider the set [itex]\Omega[/itex] of all the continuous and integrable functions [itex]f:R \to R[/itex].
Suppose we now take two subsets A and B, where:
- A is the subset of all the gaussian functions centered at the origin: [itex]\exp(-ax^2) [/itex], where a>0
- B is the set of all the even functions: [itex]f(x)=f(-x)[/itex]
It is trivial to prove that [itex]A\subset B[/itex].
However, my question is: is it possible to "quantify" how bigger the subset B is compared to A ?
How should I count the cardinality of A and B ?
My goal is computing the ratio of cardinalities: |B|/|A|. Is it possible?
Thanks.