in Set Theory & Algebra
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which of following statements are true:

1.union of countable set is countable

2.every subset of countable set is countable.

3.countable union of countable set is countable.

4 set of rational number is countable.

5. set of real number is countable

6.set of all ordered pair of non negative integer is countable.

7 product of N*N is contable
in Set Theory & Algebra
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2 Comments

  1. yes it is countable
  2. ?? 
  3. yes countable 
  4. no it is countably infinite 
  5. no it is countably infinite 
  6. yes it is  countable 
  7. yes it is  countable.
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Plz have a look at my answer..
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1 Answer

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Countable is associated with both countably finite or countably infinite..Uncountable is referred to uncountably infinite..Let us see one by one :

1) Union of countable set means it may be infinite union..Hence there is a possiblity of uncountably infinite set generation and hence making it as uncountable..

2) It is countably infinite as subset of countably infinite set will be countably infinite only..Hence it is countable..

3) Countable union of countable set is countable although it can be countably infinite..

4) Set of rational number is countably infinite [Reference : http://www.homeschoolmath.net/teaching/rational-numbers-countable.php] and hence countable..

5) Set of real numbers is uncountably infinite and hence uncountable the reason being no matter how small the interval be we can get infinite number of real numbers..Thus this set is uncountable..

6) Set of all ordered pair of non negative integer means {Z+ U {0}} *  {Z+ U {0}}..So the cardinality of this ordered pair set is nothing but mapping of each term from 1st  {Z+ U {0}} to 2nd  {Z+ U {0}}..So cardinality is : (Infinite)Infinite which will be uncountably infinite thus making this set as uncountable..

7) Similar to 6) , we get uncountably infinte set thus it is also uncountable..Only difference is now 0 is excluded ..

Hope this helps..

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4 Comments

how we say union of countable set is uncountable????

let A = set of even numbers

B=  set of odd numbers

AUB = set of all natural numbers which is countably infinite
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finite union of 2 countable sets is countable

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Mind the fact that I have mentioned the possibility of infinite union..But u r taking finite union of 2 sets only..
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@Habib. I have some queries regarding folowing points.

What I have understood from the definition of countable set is each element of the set can be counted one at a time(definition from wiki). e.g if we take a real number we can still break into infinite numbers by appending numbers making it uncountable, right?

1) Union of countable sets:  You will still be able to count each element one at a time. So, countable.

2) countable

3) Countable

4) countable

5) not countable (for reason specified above)

6) I can count one at a time e.g (0,1), (0,2) ....and so on making it countably infinite.

7) Same reason as point 6.

I want you to clarify above points.
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