$“$You cannot access the library if
you don’t have a valid ID unless
you have paid subscription fee of that day$”$
we can write like this
$“$You cannot access the library if
you don’t have a valid ID and
you don't
have paid subscription fee of that day$”$
$\neg q$ if
$\neg r$ unless
s
$\neg q$ if
$\neg r$ and
$\neg s$
$\neg q$ if
$\neg r \wedge \neg s$
$(\neg r \wedge \neg s)\rightarrow \neg q$
Implication:$P\rightarrow Q\equiv \neg P\vee Q$
Contrapositive Positive$:\neg Q\rightarrow \neg P\equiv \neg(\neg Q)\vee \neg P\equiv Q\vee \neg P\equiv\neg P\vee Q$
$Q$ if $P\equiv P\rightarrow Q$
$(\neg r \wedge \neg s)\rightarrow \neg q\equiv \neg(\neg q)\rightarrow\neg(\neg r \wedge \neg s)\equiv q\rightarrow(r\vee s)$