# Can the complement of a non-recursive language be RE

I’ve got a problem where I need to check the validity (i.e to say whether it’s true or false) of the following statement:

Complement of a non-recursive language can NEVER be recognized by any turing machine.

How I’ve thought of this is that, if a language \$mathcal{L}\$ is non-recursive, there is no turing machine that accepts the language. That is, there’s no TM that halts for every string in \$mathcal{L}\$. But there could be a TM \$M_1\$ that halts for some of the strings in \$mathcal{L}\$.

Now suppose, the proposition is \$ false\$. So there could be a TM \${M_2}\$ that recognizes the complement of \$mathcal{L}\$. So \${M_2}\$ halts and accepts every string that is NOT in \$mathcal{L}\$ and may or may not halt for strings in \$mathcal{L}\$.

Intuitively, It appears that \${M_1}\$ and \${M_2}\$ could be same, which makes my assumption \$ true\$. That is the the proposition is \$ false\$.

But I’m not certain about the arguments I’ve made (as the equivalence of \$M_1\$ and \$M_2\$ is based on intuition). Can someone verify whether I’m correct or correct me otherwise.