h\ \ reach of the following is true or false.\ is countably infinite, then

A

is infinite. is countably infinite, then

A

is countable. is uncountable, then

A

is not countably infinite. for some

kinN

, then

A

is not countable. ch of the following sets is countably infinite.\ of all natural numbers that are multiple of 5 of all integers that are multiples of 5

=N

\ athbb

(Z)/(/)(/)/(/)()/(/)\

ge

-(10)/(/)

,6

\ mathbb

(Z)/(/^(^()))m

lequiv

(2)/(/)text((mod3)/(/))/(/)

\ of Theorem 9.10.\ be sets. If

A

is infinite and

AsubeB

, then

B

is infinite.\ proof of Theorem 9.15 by proving the following:\ ountably infinite set and

x!inA

. If

f:N->A

is a bijection, then

g

is a bijection, where\

g(n)={(x if n=1),(f(n-1) if n>1.):}

\ 9.16.\ atably infinite set and

B

is a finite set, then

A\\\\cup B

is a countably infinite set.

d(B)=n

and use a proof by induction on

n

. Theorem 9.15 is the basis step. proof of Theorem 9.17 by proving the following: be disjoint countably infinite sets and let

f:N->A

and

g:N->B

be bijections. De

N}athbb{Z}\ \ge10\}\)6}mathbb{Z}\mlequiv2\text{(mod3)\}\)ofTheorem9.10.besets.IfAisinfiniteandAB,thenBisinfinite.proofofTheorem9.15byprovingthefollowing:uuntablyinfinitesetandx/A.Iff:NAisabijection,thengisabijection,wheg(n)={xf(n1)ifn=1ifn>1. 9.16. atably infinite set and B is a finite set, then AB is a countably infinite set. d(B)=n and use a proof by induction on n. Theorem 9.15 is the basis step. proof of Theorem 9.17 by proving the following: be disjoint countably infinite sets and let f:NA and g:NB be bijections. D N}athbb{Z}\ \ge10\}\)6}mathbb{Z}\mlequiv2\text{(mod3)\}\)ofTheorem9.10.besets.IfAisinfiniteandAB,thenBisinfinite.proofofTheorem9.15byprovingthefollowing:uuntablyinfinitesetandx/A.Iff:NAisabijection,thengisabijection,wheg(n)={xf(n1)ifn=1ifn>1. 9.16. atably infinite set and B is a finite set, then AB is a countably infinite set. d(B)=n and use a proof by induction on n. Theorem 9.15 is the basis step. proof of Theorem 9.17 by proving the following: be disjoint countably infinite sets and let f:NA and g:NB be bijections. D