Problem $3.$ Solve the logic puzzle below using logical formalism.

On an island, there are two kinds of robots. One kind, type H $($H for honest$),$ is programmed to always

tell the truth. The other kind, type L $($L for liar$),$ always lies. You encounter two robots R$1$ and R$2.$ What

are the types of R$1$ and R$2$ if R$1$ says $$R$2$ is type H$$ and R$2$ says $$The two of us are opposite types$?$

$$ Let p be $$R$1$ is type H$$ and let$$s assume p is true.

$$ If p is true, then what R$1$ says $($R$2$ is type H$)$ has to be true, and let$$s call this proposition q$.$

$$ If q is true, then what R$2$ says $($The two of us are opposite types$)$ has to be true.

$$ But we already have p and q above to be true saying that both R$1$ and R$2$ are type H$.$ We have a

contradiction! So$,$ our initial assumption that p is true must be wrong.

$$ Now, let$$s start with opposite, that p is false $($or $$ is true$),$ meaning $$R$1$ is type L$.$

$$ If $$R$1$ is type L$,$ then what it says $($R$2$ is type H$)$ must be a lie and therefore R$2$ must be type L

too.

$$ If R$2$ is also type L$,$ what it says $($The two of us are opposite types$)$ has to be a lie, meaning that

the two robots has to be the same type. And this now confirms what we inferred with both being

type L$.$ No contradictions, everything checks out, so$,$ both R$1$ and R$2$ are type L$.$