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TIFR-2014-Maths-B-10

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How many maps $\emptyset:\mathbb{N}\cup \left\{0\right\}\rightarrow \mathbb{N} \cup \left\{0\right\}$ are there, with the property that $\emptyset(ab)=\emptyset(a)+\emptyset(b)$, for all $a, b \in \mathbb{N} \cup \left\{0\right\}$? 

  1. None 
  2. Finitely many 
  3. Countably many
  4. Uncountably many
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