# Doubt regarding the implications of a 2-SAT constraint

Consider an example 2-SAT instance with the constraint (x1​ ∨ x2)​. This CNF has these two implications:

¬x1​→x2​ and ¬x2​→x1​.

"They actually mean, if x1​ is false then x2​ must be true, and if x2​ is false then x1 must be true, respectively. Any other case would make the 2-sat problem unsatisfiable."

My doubt is regarding this statement. I can understand the above written two implications, but I don't understand how only those two values of x1 and x2 make this 2-SAT instance satisfiable. Clearly setting both x1 and x2 to true can also satisfy this constraint. I don't see that case being captured in the two implications written above.

Can someone explain this to me? Apologies if this is a stupid question, I haven't formally studied logic.

$$A\rightarrow B$$ doesn't necessarily mean that $$B\rightarrow A$$. In your example, "$$x_1$$ is false then $$x_2$$ must be true" doesn't imply that if $$x_2$$ is true, $$x_1$$ must be false. Therefore, setting both $$x_1$$ and $$x_2$$ to true would not contradict the statement.
Also, as a side note, $$\lnot x_1\rightarrow x_2$$ is logically equivalent to $$\lnot x_2 \rightarrow x_1$$, so the two implications you described are identical.