Question

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

• A. a contradiction
• B. a tautology
• C. logically equivalent to $p\wedge (r\vee p)$
• D. logically equivalent to $p\to q$

Answer 1

The correct option is B a tautology

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

Since the truth value of the given preposition is always true, it is a tautology.