mathematics

Question

Comments

1. 2ⁿ

2. n²

3. 1

Answer 1

Let us discuss one by one :

I)  f : A --> B s.t    A  ∈  N  and B ∈  {0,1}.

We know when cardinality of A is m and that of B is n , then number of functions from A to B = n^m

So in this case no of functions = 2^|N|  

We know the set of natural  numbers is countably infinite and the second point is under countable objects , we have 2 types : a) Countably finite    b) Countably infinite 

So if a set is countably infinite also then it is also considered to be countable.

Now coming to this ,hence cardinality of set of natural numbers is countably infinite.But by Cantor's Theorem :

If  N is countably infinite then 2^N is uncountably infinite.

Hence here the number 2^|N|  is uncountably infinite and hence uncountable.So I) is false.

II)  Given ,  f : A --> B where A ∈ {0,1} and B ∈ N.

So number of such functions =  N²  

Now N is countably infinite and '2' is countably finite so the result is going to be countably infinite since here the Cantor's theorem condition is not satisfied here the base is countably infinite whereas in previous case we had exponent as countably infinite.Also by mathematical notion , 

2^N   >>>  N²  (meaning much greater)

Hence II) is countably infinite and hence countable as mentioned earlier.

III) The largest subset of a set is the set itself .So in this case is the set of natural numbers which is countably infinite and hence countable .So III) is also countable.

Hence  B should be the correct option.

Comments

III) point is not clear please explain.

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Largest subset of A ={a,b} is {a,b} itself. which is 1 set from 4 subset

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Suppose we have a set S = {1 , 2 , 3 , 4} .In that case longest subset will be the set S itself i.e. S = {1,2,3,4}..That is why in a set of natural numbers , the largest subset will be the set N itself i.e.

N = {1,2,3,4.............}

But natural number ,integer , upto rational number set we consider as countably infinite only.It is the set of real numbers and complex numbers which is considered to be uncountably infinite.The reason being simple : Between two points we can have any number of real numbers and complex numbers.

But this is not true for natural numbers , whole numbers , integers and rational numbers