Question

The statement $\left[\right(p\Rightarrow q)\wedge (q\Rightarrow r\left)\right]\Rightarrow (p\Rightarrow r)$ is

• A. a tautology
• B. a contradiction
• C. neither a tautology nor contradiction
• D. logically equivalent to $(p\wedge q)\vee r$

Answer 1

The correct option is A a tautology

$\begin{array}{ccccccccc}p& q& r& p\wedge q& p\Rightarrow q& q\Rightarrow r& p\Rightarrow r& (p\Rightarrow q)\wedge (q\Rightarrow r)& \left[\right(p\Rightarrow q)\wedge (q\Rightarrow r\left)\right]\Rightarrow (p\Rightarrow r)\\ T& T& T& T& T& T& T& T& T\\ T& T& F& T& T& F& F& F& T\\ T& F& T& F& F& T& T& F& T\\ T& F& F& F& F& T& F& F& T\\ F& T& T& F& T& T& T& T& T\\ F& T& F& F& T& F& T& F& T\\ F& F& T& F& T& T& T& T& T\\ F& F& F& F& T& T& T& T& T\end{array}$

Hence, from the above truth table, we can conclude that $\left[\right(p\Rightarrow q)\wedge (q\Rightarrow r\left)\right]\Rightarrow (p\Rightarrow r)$ is a tautology, as it's truth value's are always true.