Question
The contrapositive of $$p \to \left( { \sim q \to \, \sim r} \right)$$ is -
A.
$$\left( { \sim q \wedge r} \right) \to \, \sim p$$
B.
$$\left( { q \to r} \right) \to \, \sim p$$
C.
$$\left( {q \, \vee \sim r} \right) \to \, \sim p$$
D.
None of these
Answer :
$$\left( { \sim q \wedge r} \right) \to \, \sim p$$
Solution :
We know that the contropositive of $$p \to q$$ is $$ \sim q \to \, \sim p.$$ So contra positive of $$p \to \left( { \sim q \to \, \sim r} \right)$$ is
$$\eqalign{
& \sim \left( { \sim q \to \, \sim r} \right) \to \, \sim p \equiv \,\, \sim q \wedge \left[ { \sim \left( { \sim r} \right)} \right] \sim p \cr
& \because \,\, \sim \left( {p \to q} \right) \equiv p \, \wedge \sim q \equiv \,\, \sim q \wedge r \to \, \sim p \cr} $$