Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 264 — #272
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Chapter 8
Infinite Sets
These abstract issues about infinite sets rarely come up in mainstream mathematics,
and they don’t come up at all in computer science, where the focus is generally
on “countable,” and often just finite, sets. In practice, only logicians and set theorists
have to worry about collections that are “too big” to be sets. That’s part of
the reason that the 19th century mathematical community made jokes about “Cantor’s
paradise” of obscure infinities. But the challenge of reasoning correctly about
this far-out stuff led directly to the profound discoveries about the logical limits of
computation described in Section 8.2, and that really is something every computer
scientist should understand.
Problems for Section 8.1
Practice Problems
Problem 8.1.
Show that the set f0; 1g of finite binary strings is countable.
Problem 8.2.
Describe an example of two uncountable sets A and B such that there is no bijection
between A and B.
Problem 8.3.
Indicate which of the following assertions (there may be more than one) are equivalent
to
A strict N:
jAj is undefined.
A is countably infinite.
A is uncountable.
A is finite.
N surj A.
8n 2 N, jAj n.
8n 2 N, jAj n.