Propositional logic self doubt

Question

q = you can access the library

r = you have a valid ID

s = you have paid subscription fee of that day

Consider the following English sentence

“You cannot access the library if you don’t have a valid ID unless you have paid subscription fee of that day”

which of the following is the correct logical expression?

1. $q \rightarrow (r \vee s )$
2. $(q \rightarrow r) \vee s$

Comments

why do you add $s$ variable?

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Lakshman Patel RJIT edited

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Now see the answer.

Answer 1

$“$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 $P\equiv P\rightarrow Q$

$(\neg r \wedge \neg s)\rightarrow \neg q\equiv \neg(\neg q)\rightarrow\neg(\neg r \wedge \neg s)\equiv q\rightarrow(r\vee s)$

Comments

unless is or right?

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$P$ Unless $Q\equiv \neg Q\rightarrow P\equiv \neg(\neg Q)\vee P\equiv P\vee Q$

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¬q   if   ¬r     unless s

¬q   if   ¬r    and ¬s

why you have replaced 

¬r     unless   s

with 

¬r    and ¬s