Let $L_{1},L_{2},\cdot\cdot\cdot,L_{k}$ be a collection of languages over alphbet $\Sigma$ such that:
- For all $i\neq j$, $L_{i}\cap L_{j}=\phi$; i.e., no string is in two of the languages.
- $L_{1}\cup L_{2}\cup\cdot\cdot\cdot\cup L_{k} = \Sigma^{\ast}$;i.e., every string is in one of the languages.
- Each of the languages $L_{i}$, for $i=1,2,\cdot\cdot\cdot,k$ is recursively enumerable.
Prove that each of the languages is therefore recursive.
