Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 265 — #273
8.4. Does All This Really Work? 265
9n 2 N: jAj n.
9n 2 N: jAj < n.
Problem 8.4.
Prove that if there is a total injective (Π1 out; 1 in) relation from S to N, then
S is countable.
Problem 8.5.
Prove that if S is an infinite set, then pow S is uncountable.
Problem 8.6.
Let A to be some infinite set and B to be some countable set. We know from
Lemma 8.1.7 that
A bij .A [ fb 0 g/
for any element b 0 2 B. An easy induction implies that
A bij .A [ fb 0 ; b 1 ; : : : ; b n g/ (8.8)
for any finite subset fb 0 ; b 1 ; : : : ; b n g B.
Students sometimes think that (8.8) shows that A bij .A [ B/. Now it’s true that
A bij .A [ B/ for all such A and B for any countable set B (Problem 8.13), but the
facts above do not prove it.
To explain this, let’s say that a predicate P.C / is finitely discontinuous when
P.A [ F / is true for every finite subset F B, but P.A [ B/ is false. The hole
in the claim that (8.8) implies A bij .A [ B/ is the assumption (without proof) that
the predicate
P 0 .C / WWD ŒA bij C
is not finitely discontinuous. This assumption about P 0 is correct, but it’s not completely
obvious and takes some proving.
To illustrate this point, let A be the nonnegative integers and B be the nonnegative
rational numbers, and remember that both A and B are countably infinite.
Some of the predicates P.C / below are finitely discontinuous and some are not.
Indicate which is which.
1. C is finite.