I get confused when i think about logical equivalence between conditional statements. For example saying that
∼(P⇒Q)=P∧∼Q.
If there is variables involved then the statement on the left says that there exists some value of that variable for which P does not imply Q. The statement on the right P and not Q is true for all values of that variable, This is what i understand but i think im wrong because these statements are logically equivalent and are meant to say the same thing about P and Q.
I can understand if they did not include variables but not if they do can someone help explain, thanks.
The way you are trying to understand equivalence is slightly correct, but mostly not: first one is saying $P$ does not imply $Q$. In a truth-table, if you notice, a conditional is true in two and only two cases: either $P$ is false or $P$ is true and $Q$ is true. Therefore, to say $P$ does not imply $Q$ is to say: $P$ is true and $Q$ is false.
Now notice, $P$ and not $Q$ is true exactly when both $P$ is true and $Q$ is false. Did you notice anything similar between this sentence and the last sentence of the first paragraph?
This similarity means $\lnot (P \implies Q)$ and $P \land \lnot Q$ have the same truth-table. Therefore, they are uquivalent.
That said, falisty and truth of a propositional statement depends on the variables with respect to the assignment of truth values to those variables.