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Test by Bikram | Mock GATE | Test 4 | Question: 17

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Let  the set of all natural numbers be $N$. Which of the following are countable sets?

  1.  The set of all functions from $N$ to $\{0, 1\}$
  2.  The set of all functions from $\{0, 1\}$ to $N$
  3.  The largest subset of $N$
  1.    I, II, and III
  2.    I and II only
  3.    II and III only
  4.    I and III only
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The set of all functions from $Q$ to $\left \{ 0,1 \right \}$are uncountable 

reference:https://math.stackexchange.com/questions/129851/show-that-the-set-of-functions-mathbbn-to-0-1-is-not-countable

Set of all natural numbers is countable because there exists one to one correspondence to each number of this set.

There is one more proof which says Every subset of countable set is either finite or countable, that's why the third statement is correct

option $C$ will be the answer...

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