Is infinite union of finite sets countable?
0
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I wasn't sure of whether infinite union of finite sets is countable?
My logic :
if $$ A_{i} $$ is a finite subset of any countable set $B$ then:-
$$ bigcup_{i=1}^{infty }A_{i} =B $$
And we know $B$ is countable so does it prove infinite union of finite sets is countable?
real-analysis cardinals
edited Dec 21 '18 at 12:18
mrtaurho
6,10771641
asked Dec 21 '18 at 12:07
Amisha BansalAmisha Bansal
64
$endgroup$
add a comment |
0
$begingroup$
I wasn't sure of whether infinite union of finite sets is countable?
My logic :
if $$ A_{i} $$ is a finite subset of any countable set $B$ then:-
$$ bigcup_{i=1}^{infty }A_{i} =B $$
And we know $B$ is countable so does it prove infinite union of finite sets is countable?
real-analysis cardinals
edited Dec 21 '18 at 12:18
mrtaurho
6,10771641
asked Dec 21 '18 at 12:07
Amisha BansalAmisha Bansal
64
$endgroup$
7
$begingroup$
Any set at all is an infinite union of singletons.
$endgroup$
– Ittay Weiss
Dec 21 '18 at 12:09
1
$begingroup$
A countable union of finite sets is countable.
$endgroup$
– William Elliot
Dec 21 '18 at 12:19
2
$begingroup$
@William: Not unless you assume the axiom of choice.
$endgroup$
– Asaf Karagila♦
Dec 21 '18 at 12:19
$begingroup$
@WilliamElliot If it isn't a countable union then it can result in countable or uncountable set?
$endgroup$
– Amisha Bansal
Dec 21 '18 at 12:20
3
$begingroup$
Yes, of course. The set of all real numbers is an uncountable union of singleton sets. On the other hand, if many of the sets have non-empty intersection, an uncountable union may still be countable.
$endgroup$
– user247327
Dec 21 '18 at 12:27
add a comment |
0
0
0
$begingroup$
I wasn't sure of whether infinite union of finite sets is countable?
My logic :
if $$ A_{i} $$ is a finite subset of any countable set $B$ then:-
$$ bigcup_{i=1}^{infty }A_{i} =B $$
And we know $B$ is countable so does it prove infinite union of finite sets is countable?
real-analysis cardinals
edited Dec 21 '18 at 12:18
mrtaurho
6,10771641
asked Dec 21 '18 at 12:07
Amisha BansalAmisha Bansal
64
$endgroup$
I wasn't sure of whether infinite union of finite sets is countable?
My logic :
if $$ A_{i} $$ is a finite subset of any countable set $B$ then:-
$$ bigcup_{i=1}^{infty }A_{i} =B $$
And we know $B$ is countable so does it prove infinite union of finite sets is countable?
real-analysis cardinals
real-analysis cardinals
edited Dec 21 '18 at 12:18
mrtaurho
6,10771641
asked Dec 21 '18 at 12:07
Amisha BansalAmisha Bansal
64
edited Dec 21 '18 at 12:18
mrtaurho
6,10771641
asked Dec 21 '18 at 12:07
Amisha BansalAmisha Bansal
64
edited Dec 21 '18 at 12:18
mrtaurho
6,10771641
edited Dec 21 '18 at 12:18
mrtaurho
6,10771641
edited Dec 21 '18 at 12:18
mrtaurho
6,10771641
6,10771641
asked Dec 21 '18 at 12:07
Amisha BansalAmisha Bansal
64
asked Dec 21 '18 at 12:07
Amisha BansalAmisha Bansal
64
asked Dec 21 '18 at 12:07
Amisha BansalAmisha Bansal
64
64
7
$begingroup$
Any set at all is an infinite union of singletons.
$endgroup$
– Ittay Weiss
Dec 21 '18 at 12:09
1
$begingroup$
A countable union of finite sets is countable.
$endgroup$
– William Elliot
Dec 21 '18 at 12:19
2
$begingroup$
@William: Not unless you assume the axiom of choice.
$endgroup$
– Asaf Karagila♦
Dec 21 '18 at 12:19
$begingroup$
@WilliamElliot If it isn't a countable union then it can result in countable or uncountable set?
$endgroup$
– Amisha Bansal
Dec 21 '18 at 12:20
3
$begingroup$
Yes, of course. The set of all real numbers is an uncountable union of singleton sets. On the other hand, if many of the sets have non-empty intersection, an uncountable union may still be countable.
$endgroup$
– user247327
Dec 21 '18 at 12:27
add a comment |
7
$begingroup$
Any set at all is an infinite union of singletons.
$endgroup$
– Ittay Weiss
Dec 21 '18 at 12:09
1
$begingroup$
A countable union of finite sets is countable.
$endgroup$
– William Elliot
Dec 21 '18 at 12:19
2
$begingroup$
@William: Not unless you assume the axiom of choice.
$endgroup$
– Asaf Karagila♦
Dec 21 '18 at 12:19
$begingroup$
@WilliamElliot If it isn't a countable union then it can result in countable or uncountable set?
$endgroup$
– Amisha Bansal
Dec 21 '18 at 12:20
3
$begingroup$
Yes, of course. The set of all real numbers is an uncountable union of singleton sets. On the other hand, if many of the sets have non-empty intersection, an uncountable union may still be countable.
$endgroup$
– user247327
Dec 21 '18 at 12:27
7
7
$begingroup$
Any set at all is an infinite union of singletons.
$endgroup$
– Ittay Weiss
Dec 21 '18 at 12:09
$begingroup$
Any set at all is an infinite union of singletons.
$endgroup$
– Ittay Weiss
Dec 21 '18 at 12:09
1
1
$begingroup$
A countable union of finite sets is countable.
$endgroup$
– William Elliot
Dec 21 '18 at 12:19
$begingroup$
A countable union of finite sets is countable.
$endgroup$
– William Elliot
Dec 21 '18 at 12:19
2
2
$begingroup$
@William: Not unless you assume the axiom of choice.
$endgroup$
– Asaf Karagila♦
Dec 21 '18 at 12:19
$begingroup$
@William: Not unless you assume the axiom of choice.
$endgroup$
– Asaf Karagila♦
Dec 21 '18 at 12:19
$begingroup$
@WilliamElliot If it isn't a countable union then it can result in countable or uncountable set?
$endgroup$
– Amisha Bansal
Dec 21 '18 at 12:20
$begingroup$
@WilliamElliot If it isn't a countable union then it can result in countable or uncountable set?
$endgroup$
– Amisha Bansal
Dec 21 '18 at 12:20
3
3
$begingroup$
Yes, of course. The set of all real numbers is an uncountable union of singleton sets. On the other hand, if many of the sets have non-empty intersection, an uncountable union may still be countable.
$endgroup$
– user247327
Dec 21 '18 at 12:27
$begingroup$
Yes, of course. The set of all real numbers is an uncountable union of singleton sets. On the other hand, if many of the sets have non-empty intersection, an uncountable union may still be countable.
$endgroup$
– user247327
Dec 21 '18 at 12:27
add a comment |
0
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7
$begingroup$
Any set at all is an infinite union of singletons.
$endgroup$
– Ittay Weiss
Dec 21 '18 at 12:09
1
$begingroup$
A countable union of finite sets is countable.
$endgroup$
– William Elliot
Dec 21 '18 at 12:19
2
$begingroup$
@William: Not unless you assume the axiom of choice.
$endgroup$
– Asaf Karagila♦
Dec 21 '18 at 12:19
$begingroup$
@WilliamElliot If it isn't a countable union then it can result in countable or uncountable set?
$endgroup$
– Amisha Bansal
Dec 21 '18 at 12:20
3
$begingroup$
Yes, of course. The set of all real numbers is an uncountable union of singleton sets. On the other hand, if many of the sets have non-empty intersection, an uncountable union may still be countable.
$endgroup$
– user247327
Dec 21 '18 at 12:27