Question

If statement $(p\to q)\to (q\to r)$ is false, then truth values of statements $p$, $q$ and $r$ respectively can be:
(a) FTF
(b) TTT
(c) FFF
(d) FTT

Answer 1

We will look at the truth table for the logical implication to solve this question. In the given statement, there are three implications. According to the brackets placed and the given truth value of the statement, we will deduce the truth values for all three statements $p$, $q$, and $r$. We will consider the truth values for $(p\to q)$ and $(q\to r)$ in such a manner that the given statement should have the truth value to be false.

Complete step-by-step solution:
The truth table for logical implication is as follows:

$a$	$b$	$a\to b$
T	T	T
T	F	F
F	T	T
F	F	T

where $a$ and $b$ are two statements. Now, the statement given to us is $(p\to q)\to (q\to r)$, and its truth value is false. According to the truth table for logical implication, the only possibility for an implication to be false is the following:
$(p\to q)$ is true and $(q\to r)$ is false.
Now, if $(q\to r)$ is false, then using the same reasoning as above, we can say that the truth value of statement $q$ is true and that of statement $r$ is false. The only option which has the truth value of statement $q$ as true and the truth value of statement $r$ as false is an option (a).
So, the correct answer is an option (a).
Note: Referring to a correct truth table is very crucial for such questions. Understanding the precedence of statements by the use of brackets is also important. These are the areas where mistakes happen often. Eliminating options from the given choices in a question is another method of solving multiple-choice questions.