Examples
- If d(A) exists for some set A, then for the complement set we have d(Ac) = 1 − d(A).
- Obviously, d(N) = 1.
- For any finite set F of positive integers, d(F) = 0.
- If is the set of all squares, then d(A) = 0.
- If is the set of all even numbers, then d(A) = 0.5 . Similarly, for any arithmetical progression we get d(A) = 1/a.
- For the set P of all primes we get from the prime number theorem d(P) = 0.
- The set of all square-free integers has density
- The density of the set of abundant numbers is known to be between 0.2474 and 0.2480.
- The set of numbers whose binary expansion contains an odd number of digits is an example of a set which does not have an asymptotic density, since the upper density of this set is
-
- whereas its lower density is
- Consider an equidistributed sequence in and define a monotone family of sets :
- Then, by definition, for all .
Read more about this topic: Natural Density
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