Proof language Turing-decidable

Question

Hello I should proof that: Let L_1, L_2, L_3 be Turing-recognizable language such that L_1 U L_2 U L_3 = {a, b}* and (L_i [intersection] L_j = [empty-set]) for any 1<=i<j<=3. Show that L_1 is Turing-decidable. I have no idea.

Answer 1

First of all, this question is not appropriate to Stack overflow, as it is a math question.

Second of all the answer is, to show that L_1 is Turing-decidable, given that we already know it's Turing-recognizable, you need to show that the complement, L_1^c inside {a,b}^* is Turing recognizable. But this complement is exactly L_2 U L_3, by assumption. And the union of two Turing recognizable languages is also turing recognizable.