Density: Difference between revisions
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Let X be a finite set. The usual definition of the density of a subset Y of X is |Y|/|X|, that is, the size of Y divided by the size of X. In particular, if <math>\mathcal{A}</math> is a subset of <math>[3]^n</math> then its density is <math>3^{-n}|\mathcal{A}.</math> | Let X be a finite set. The usual definition of the density of a subset Y of X is |Y|/|X|, that is, the size of Y divided by the size of X. In particular, if <math>\mathcal{A}</math> is a subset of <math>[3]^n</math> then its density is <math>3^{-n}|\mathcal{A}|.</math> | ||
One speaks loosely of a set <math>\mathcal{A}\subset[3]^n</math> being ''dense'' if its density <math>\delta</math> is bounded below by a positive constant that is independent of n. Strictly speaking, this definition applies to sequences of sets with n tending to infinity, but it is a very useful way of talking. | One speaks loosely of a set <math>\mathcal{A}\subset[3]^n</math> being ''dense'' if its density <math>\delta</math> is bounded below by a positive constant that is independent of n. Strictly speaking, this definition applies to sequences of sets with n tending to infinity, but it is a very useful way of talking. | ||
Sometimes it is helpful to consider other probability measures on <math>[3]^n,</math> such as [[equal-slices_measure|equal-slices density]]. Then the words "density" and "dense" have obviously analogous uses. | Sometimes it is helpful to consider other probability measures on <math>[3]^n,</math> such as [[equal-slices_measure|equal-slices density]]. Then the words "density" and "dense" have obviously analogous uses. | ||
Latest revision as of 04:33, 23 February 2009
Let X be a finite set. The usual definition of the density of a subset Y of X is |Y|/|X|, that is, the size of Y divided by the size of X. In particular, if [math]\displaystyle{ \mathcal{A} }[/math] is a subset of [math]\displaystyle{ [3]^n }[/math] then its density is [math]\displaystyle{ 3^{-n}|\mathcal{A}|. }[/math]
One speaks loosely of a set [math]\displaystyle{ \mathcal{A}\subset[3]^n }[/math] being dense if its density [math]\displaystyle{ \delta }[/math] is bounded below by a positive constant that is independent of n. Strictly speaking, this definition applies to sequences of sets with n tending to infinity, but it is a very useful way of talking.
Sometimes it is helpful to consider other probability measures on [math]\displaystyle{ [3]^n, }[/math] such as equal-slices density. Then the words "density" and "dense" have obviously analogous uses.
