I know that given two subsets of the Naturals A and B, if the natural density of A equals some non-zero real number a, and the natural density of B is zero, then the natural density of the symmetric difference of A and B is still a. But what are the properties of just set difference with respect to natural density? That is, suppose we have the sets A and B, with respective natural densities as above. Would the natural density of the set difference A-B still be a?
2026-05-04 09:02:49.1777885369
Does the natural (asymptotic) density of a set A change if a subset of A with natural density zero is subtracted from A?
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$A\setminus B=A\Delta (B\cap A)$, and $B\cap A$ still has density zero since $B$ does and $B\cap A\subseteq B$.