Q. The statement $p \rightarrow \left(q\,\rightarrow \,p\right) $ is equivalent to:
Solution:
p q $q \rightarrow p$ $p \rightarrow\left(q \rightarrow p\right)$ $p\,\vee\,q$ $p \rightarrow \left(p\,\vee \,q\right)$ T T T T T T T F F T T T F T T T T T F F T T F T
Since truth value of $p \rightarrow \left(q\,\rightarrow \,p\right)$ and $p \rightarrow \left(p\,\vee\,q\right)$ are same, hence $p \rightarrow \left(q\,\rightarrow \,p\right)$ is equivalent to $p \rightarrow \left(p\,\vee \,q\right).$
p | q | $q \rightarrow p$ | $p \rightarrow\left(q \rightarrow p\right)$ | $p\,\vee\,q$ | $p \rightarrow \left(p\,\vee \,q\right)$ |
---|---|---|---|---|---|
T | T | T | T | T | T |
T | F | F | T | T | T |
F | T | T | T | T | T |
F | F | T | T | F | T |