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We know a function f : A --> B is onto function iff :

                        Range(A)  = Co domain [Co domain is nothing but set B possible values]

Now for a given value of A , we will have unique and hence one value of B..

So say if we have 10 unique values of A , we will have at most 10 values of B [If we have 10 values then it would be a bijection as in that case function will be one one and onto is given in the question]

In short , as the function f : A --> B is onto , hence 

No of values in set B  <=   No of set A  [ Equality holds when the function is one - one also]

As the set A is given to be countable , so anything less than countable is countable as well..

Hence set B is also countable..

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