Is there a mapping reduction for every two language $A$ and $B$ to some language $C$? One of my friend told me that there is a language $C$ for every two languages $A$ and $B$ s.t $A \leq_{m} C$ and $B \leq_{m} C$ , he simply define two languages $A’=\{0w|w \in A\}$ and $B’=\{1w|w \in B\}$ so $C=A’ \cup B’$ and he define the function $f(w)=0w$ if $w \in A$ and and $f(w)=1w$ otherwise. I suspect this function is turing-computable. Suppose $A$ be $\overline{HP}$ which is not RE. How we can decide if $w \in A$ ? Even i ask myself can we realy prove that such a mapp-reduction always exist ?
The definition of the reduction is wrong.
We actually need to define two reductions: from $A$ to $C$ and from $B$ to $C$. The reduction from $A$ to $C$ maps $w$ to $0w$. The reduction from $B$ to $C$ maps $w$ to $1w$.