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Let A be the set of all sequences having only 0 and 1 in its range. Can an onto mapping be de ned from N to A?

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Question: Let A be the set of all sequences having only 0 and 1 in its range. Can an onto mapping be de ned from N to A? Solution: We know that each element x [0, 1] can be uniquely expressed in binary form as x = 02 a1 a2 . . ., where ai = 0 or 1 i N. We de ne a mapping f : A [0, 1] by f (a1 a2 a3 . . . an . . .) = 02 a1 a2 . . . an . . . (a1 a2 a3 . . . an . . .) A Clearly, f is onto. Since, [0, 1] is uncountable, so A is uncountable. Hence, it is not possible to de ne an onto map from N to A. Solved by ARNAB BHATTACHARJEE (http://www.facebook.com/itzArnab)

 

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