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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

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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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