Question

Identify the correct steps involved in proving that the union of a countable number of countable sets is countable. (Check all that apply.) Check All That Apply Since empty sets do not contribute any elements to unions, we can assume that none of the sets in our given countable collection of countable sets is an empty set. If there are no sets in the collection, then the union is empty and therefore countable, Otherwise let the countable sets be As, Az. Since each set As countable and nonempty, we can list its elements in a sequence as an .. again, if the set is finite, we can list its elements and the list a repeatedly to assure an infinite sequence Otherwise let the countable sets be As, Az. Since each set A countable and onemoty, we cannot list its elements in a sequence as . gain, if the set is finite, we cannot list its elements and then repeatedly to assure an infinite sequence We can put all the elements into a sequence in a system way by ...then the elements in which there are only two such pai element that we have already listed n say in which +/- 2 there is only one such and soon except that we do not stany We can put all the clements into a sequence in a systematic way by pel t hen all the elements in which 1 -3 here are only one such which there is only one such and so on except that we do not that

Answer 1

Select Option First.

Which means if all thesets are empty then union becomes empty so result in this case holds trivially.

Select Second

Because we always can make sequence of these sets and if these sets are finite then again result holds trivially.

Don't Select Third

Because it states we can not make sequence but we can do.

Select Fourth.

As in this way number of choices will be partition of a particular number in two pairs.

For 2=1+1

For 3=1+2,2+1

For 4=1+3,3+1,2+2 and so on

Don't Select Fifth

As it states there is only one pair for 3 which is 2+1 but we know there is one more 1+2 also.

Select Sixth

As it states that we can express all elements as a sequence and we can omit the repeat elements.

Select Last

From above step we conpared the whole sequence as we did for rational numbers to compare Rationals with Natural Numbers.

And finally our result is proved.