Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 252 — #260
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Chapter 8
Infinite Sets
More Countable and Uncountable Sets
Once we have a few sets we know are countable or uncountable, we can get lots
more examples using Lemma 8.1.3. In particular, we can appeal to the following
immediate corollary of the Lemma:
Corollary 8.1.15.
(a) If U is an uncountable set and A surj U , then A is uncountable.
(b) If C is a countable set and C surj A, then A is countable.
For example, now that we know that the set f0; 1g ! of infinite bit strings is uncountable,
it’s a small step to conclude that
Corollary 8.1.16. The set R of real numbers is uncountable.
To prove this, think about the infinite decimal expansion of a real number:
p
2 D 1:4142 : : : ;
5 D 5:000 : : : ;
1=10 D 0:1000 : : : ;
1=3 D 0:333 : : : ;
1=9 D 0:111 : : : ;
4 1 99 D 4:010101 : : : :
Let’s map any real number r to the infinite bit string b.r/ equal to the sequence
of bits in the decimal expansion of r, starting at the decimal point. If the decimal
expansion of r happens to contain a digit other than 0 or 1, leave b.r/ undefined.
For example,
b.5/ D 000 : : : ;
b.1=10/ D 1000 : : : ;
b.1=9/ D 111 : : : ;
b.4 1 99 / D 010101 : : :
b. p 2/; b.1=3/ are undefined: