# If $f$ reduces $L_1$ to $L$ and also $L_2$ to $L$ is $L_1=L_2$

If the same $$f$$ reduces $$L_1$$ to $$L$$ and also $$L_2$$ to $$L$$ does it imply that $$L_1=L_2$$?

My intuition says no, but I couldn't find a counterexample.

• Hint: There is a very concise way to express $L_1$ using $f$ and $L$. The same for $L_2$. Check the definition of reduction.
– user114966
Mar 22 '21 at 2:45

$$x\in L_1\iff f(x)\in L \iff x\in L_2$$
And thus $$L_1=L_2$$.
When a reduction takes place that means that the problem $$L_i$$ is just a different "point of view" of problem $$L$$. Given that you have two problems $$L_1$$ and $$L_2$$ reducible to the same $$L$$ then you could say that by solving $$L$$ you would be able to solve both. This in turn implies that they could be treated as the same problem.