# To prove $A\Rightarrow B$ is it enough to prove that $\lnot(A\Rightarrow B)$ is false with a counterexample?

I want to prove $A\Rightarrow B$ is true. If I prove that $\lnot(A\Rightarrow B)$ is false with a counter example is it enough?

• How would you prove $\neg(A\Rightarrow B)$ with a counterexample? Could you provide an example of such situation? Dec 5, 2015 at 20:47
• Do you have a specific problem in mind? I ask because the true negation of $\lnot (A \implies B)$ should be truly equivalent to $A \implies B$, and it's quite rare to prove things by exhibiting an example where it's true... Dec 5, 2015 at 20:50
• @Wojowu with $\urcorner(A\Rightarrow B)$ I mean A doesn't imply B. Dec 5, 2015 at 21:25
• How do you prove something with counterexample, counterexample usually used to disprove something like a conjecture Dec 6, 2015 at 12:50

## 1 Answer

Intuitively speaking :

• $A \Rightarrow B$ means "whenever $A$ holds than $B$ holds".

• $\neg(A \Rightarrow B)$ means "it is not true that whenever $A$ holds than $B$ holds", i.e there exist a case where $A$ hold but $B$ does not hold.

Therefore a counter-example to the universal statement $A \Rightarrow B$ is a proof of $\neg(A \Rightarrow B)$. However there is no such thing as a counter-example to $\neg(A \Rightarrow B)$ because this is not a universal statement. To disprove $\neg(A \Rightarrow B)$ you need to prove that "whenever $A$ holds than $B$ holds" which is the statement $A \Rightarrow B$.