Cardinality
Margaret M. Fleck
5 May 2010
This is a half lecture due to the makeup quiz. It discussed cardinality,
an interesting topic but which doesn’t have an obvious fixed place in the
syllabus. It’s covered at the very end of section 2.4 in Rosen.
1
The rationals and the reals
You’re familiar with three basic sets of numbers: the integers, the rationals,
and the reals. The integers are obviously discrete, in that there’s a big gap
between successive pairs of integers.
To a first approximation, the rational numbers and the real numbers seem
pretty similar. The rationals are dense in the reals: if I pick any real number
x
and a distance
δ
, there is always a rational number within distance
δ
of
x
.
Between any two real numbers, there is always a rational number.
We know that the reals and the rationals are different sets, because we’ve
shown that a few special numbers are not rational, e.g.
π
and
√
2. However,
these irrational numbers seem like isolated cases.
In fact, this intuition is
entirely wrong:
the vast majority of real numbers are irrational and the
rationals are quite a small subset of the reals.
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 Spring '10
 fleck
 Rational number, reals

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