Question

If the statement $(p\to q)\to (q\to r)$ is false, then truth values of statement $p,q,r$ respectively, can be

• A. $\text{F T F}$
• B. $\text{T F T}$
• C. $\text{T F F}$
• D. $\text{F T T}$

Answer 1

The correct option is A $\text{F T F}$

As given that $(p\to q)\to (q\to r)$ is false, it means
$p\to q$ must be true and $q\to r$ must be false.
Now, as $(q\to r)$ is false so, $q$ must be true and $r$ must be false and $p$ can be true or false.

Hence, the truth values of statement $p,q,r$ will be $\text{F T F}$ or $\text{T T F}$ respectively.

Alternate Solution:
$\begin{array}{cccccc}p& q& r& p\to q& q\to r& (p\to q)\to (q\to r)\\ T& T& T& T& T& T\\ T& T& F& T& F& F\\ T& F& T& F& T& T\\ T& F& F& F& T& T\\ F& T& T& T& T& T\\ F& T& F& T& F& F\\ F& F& T& T& T& T\\ F& F& F& T& T& T\end{array}$

Clearly, from truth table
$(p\to q)\to (q\to r)$ is false when truth values of statement $p,q,r$ are $\text{F T F}$ or $\text{T T F}$ respectively.