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2 votes
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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. $(q \rightarrow r) \vee s$
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$“$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 $P\equiv P\rightarrow Q$

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

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