bijection between unit interval and unit
square∗
pahio†
2013-03-22 4:01:42
The real numbers in the open unit interval I = (0, 1) can be uniquely
represented by their decimal expansions, when these must not end in an infinite
string of 9’s. Correspondingly, the elements of the open unit square I ×I are
represented by the pairs of such decimal expansions.
Let
P := (0.x1 x2 x3 . . . , 0.y1 y2 y3 . . .)
be such a pair representing an arbitrary point in I ×I and let
p := 0.x1 y1 x2 y2 x3 y3 . . .
Then it’s apparent that
P 7→ p
(1)
is an injective mapping from I ×I to I. Thus
|I ×I| ≤ |I|.
But since I×I contains more than one horizontal open segment equally long as
I (and accordingly there is a natural injection from I to I ×I), we must have
also
|I ×I| ≥ |I|.
The conclusion is that
|I ×I| = |I|,
i.e. that the sets I×I and I have equal cardinalities, and the Schröder–Bernstein
theorem even garantees a bijection between the sets.
∗ hBijectionBetweenUnitIntervalAndUnitSquarei
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1
Remark. Georg Cantor utilised continued fractions for constructing such a
bijection between the unit interval and the unit square; cf. e.g. this MAA article.
Since the mapping g : I → R defined by
π
g(x) = tan πx −
2
is bijective, we can conclude that the sets R and R×R, i.e. the set of the points
of a line and the set of the points of a plane, have the same cardinalities. This
common cardinality is 2ℵ0 .
References
[1] Michael B. Williams: Cardinality (2011). Available here.
2