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) \[\left( q\wedge r \right)\to \sim p\]
(c) \[\left( q\wedge \sim r \right)\to \sim p\]
(d) None of these

Answer 1

Hint: For solving this problem we use a simple definition of discrete mathematics. Contrapositive is nothing but negating the given statement which is denoted by \['\sim '\] and the symbol \['\to '\] is used for conditional statements. We use simple transformations listed below to solve this problem
\[\begin{align}
  & \sim \left( p\to q \right)=\sim q\to \sim p \\
 & \sim q\to \sim p=q\vee \sim p \\
 & \sim \left( q\vee p \right)=\sim q\wedge \sim p \\
 & \sim \left( \sim p \right)=p \\
\end{align}\]
Here, the symbols \['\wedge ,\vee '\] are denoted for ‘and’, ‘or’ statements respectively.

Complete step by step answer:
Let us take the given statement as
\[x=p\to \left( \sim q\to \sim r \right)\]
We know that the contrapositive of a statement is nothing but applying the negation to the whole statement.
By applying the negation to given statement we will get
\[\Rightarrow x=\sim \left( p\to \left( \sim q\to \sim r \right) \right)\]
We know that the negation of a conditional statement is given as \[\sim \left( p\to q \right)=\sim q\to \sim p\].

By applying this formula to above equation we will get
\[\Rightarrow x=\sim \left( \sim q\to \sim r \right)\to \sim p\]
We know that the conditional statement of two negation statements is given as \[\sim q\to \sim p=q\vee \sim p\]
By applying this formula to above equation we will get
\[\Rightarrow x=\sim \left( q\vee \sim r \right)\to \sim p\]
By taking the negation of first statement in above equation inside then the ‘or’ symbol is converted to ‘and’ symbol negation is applied to both inside statements that is \[\sim \left( q\vee p \right)=\sim q\wedge \sim p\]
So, the above equation can be modified as
\[\Rightarrow x=\left( \sim q\wedge \sim \left( \sim r \right) \right)\to \sim p\]
We know that the negation of the negation statement given the original statement that is \[\sim \left( \sim p \right)=p\].
By applying this formula to above statement we will get
\[\Rightarrow x=\left( \sim q\wedge r \right)\to \sim p\]
Therefore, the answer is \[\left( \sim q\wedge r \right)\to \sim p\].

So, the correct answer is “Option a”.

Note: Students may make mistakes in applying the discrete mathematics formulas that are mentioned above. In some formulas there will be interchanging of positions of statements after applying the negation to conditional statements. That is
\[\sim \left( p\to q \right)=\sim q\to \sim p\]
Here, students may miss in interchanging and takes the formula as
\[\sim \left( p\to q \right)=\sim p\to \sim q\]
This will be wrong. So the application of formula needs to be taken care of.