Question

The statement $(p\to (q\to p\left)\right)\to (p\to (p\vee q\left)\right)$ is:

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

Answer 1

The correct option is D

a tautology

Explanation for the correct option:

Given, $(p\to (q\to p\left)\right)\to (p\to (p\vee q\left)\right)$

Make a truth table

$\begin{array}{|ccccccc|}\hline p& q& q\to p& p\to \left(q\to p\right)& p\vee q& p\to \left(p\vee q\right)& (p\to \left(q\to p\right))\to (p\to \left(p\vee q\right))\\ T& T& T& T& T& T& T\\ T& F& T& T& T& T& T\\ F& T& F& T& T& T& T\\ F& F& T& T& F& T& T\\ \hline\end{array}$

Thus, $(p\to (q\to p\left)\right)\to (p\to (p\vee q\left)\right)$ is a tautology.

Hence, option (D) is the correct answer.