Question

The contrapositive of the sentence \[p \to q\] is equivalent to
A) \[p \to \neg q\]
B) \[q \to \neg p\]
C) \[q \to p\]
D) \[\neg p \to \neg q\]
E) \[\neg q \to \neg p\]

Answer 1

Hint:
Here we have to use the concept of the converse and inverse statement to get the contrapositive of the given sentence. We will first find the converse of the given sentence and then we will find the inverse of the converse sentence to get the contrapositive of the given sentence.

Complete step by step solution:
We know that the given sentence is \[p \to q\].
Now we will find the converse of the given sentence.
Converse of the sentence \[p \to q\] is \[q \to p\].
Now we have to find the value of the inverse of the converse of \[q \to p\].
So the inverse of \[q \to p\] is \[\neg q \to \neg p\].
We know that the contrapositive sentence is the converse and inverse of the given sentence. So this inverse sentence is the contrapositive sentence of the given equation.
 Hence, \[\neg q \to \neg p\] is the contrapositive of the sentence \[p \to q\].

So, we can conclude option E is the correct option.

Note:
Here we need to know the meaning of converse. The converse of a sentence is the process of interchanging the roles of the hypothesis and conclusion of the original sentence. We can say that the hypothesis will become the conclusion and the conclusion will become the hypothesis. Also, the contrapositive sentence is the combination of the converse and inverse of the given sentence. Firstly we have to find the converse and then its inverse to get the contrapositive of the sentence and the other way round.
In the inverse of a sentence, the symbol \[\neg p\] is read as “not p” while \[\neg q\] is read as “not q”