Question

Let $p\equiv$ "You apply for a driving licence.", $q\equiv$ "You should have a ration card." and $r\equiv$ "You should have a passport."
What can be concluded from "To apply for a driving licence, you should have a ration card or a passport"

• A. $(\sim q\wedge r)\Rightarrow \sim p$
• B. $(\sim q\wedge \sim r)\Rightarrow \sim p$
• C. $(\sim q\wedge \sim r)\Rightarrow p$
• D. $(q\wedge \sim r)\Rightarrow \sim p$

Answer 1

The correct option is B $(\sim q\wedge \sim r)\Rightarrow \sim p$

"To apply for a driving licence, you should have a ration card or a passport"
can be rewritten as
"If you apply for a driving licence, then you should have a ration card or a passport."
which is given by $p\Rightarrow (q\vee r)$
it's contrapositive is given by $\sim (q\vee r)\Rightarrow \sim p$
$(\sim q\wedge \sim r)\Rightarrow \sim p$