Reading my introduction to discrete math book comes with an introduction to logic. I understand the truth table for $p \rightarrow q$ but I am getting confused why $q\:unless\:\neg\:p$ is a similar statement because I'm not sure how to read it, specifically the word unless. How do these two compound propositions mean the same thing and can you give an example translated to English? Perhaps knowing how unless affects the statement can help me figure out the rest.
You need to look at $p$, not $\lnot p$. In your case, if $q$ is false and $\lnot p$ is false (i.e. $p$ is true), $q$ unless $\lnot p$ is false, so in order of ($p$, $q$, statement) it is (true, false, false) which is the same as $p \rightarrow q$
They are different in English but the same in math. If the politician is not elected ($p = F$), and he lowered tax ($q = T$), he will lower tax if he is elected is true while he will lower tax unless he is not elected sounds like a false. The confusion comes because in mathematics, given the conditions above, (he will lower tax unless he is elected) is true.
I think a better way to explain $q$ unless $\lnot p$ is "$q$ unless $\lnot p$, but if $\lnot p$ really happens then $q$ doesn't matter". (like what @skyking said)
Actually I believe Are these two statements equivalent? is the same question.