#MathsInMinutes Day 27: Dense sets

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Density is a property that describes relationships between sets and their subsets when there is a notion of distance between elements of the set. It provides a way of assessing the relative 'size' of different infinite sets that is different from counting the elements. For instance, one way to make mathematical sense of the idea that the rational numbers are a 'very big' set, is that they are dense within a specific subset, in this case the real numbers, which are 'very big' themselves.

A set X is said to be dense in another set Y, if X is a subset of Y, and any point in X is either an element of Y, or arbitrarily close to one: for any point in Y we can choose any distance d greater than 0 and find a point in X within distance d of that point.

To prove that the rationals are dense in the reals, for example, we select a distance d and a real number y, then prove that there is always a rational number x within d of y, which can be done by truncating the decimal expansion of y.

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