Question

$Iftheproposition\neg p\Rightarrow qistrue,$
then the truth value of the proposition
$\neg P\vee (p\Rightarrow q)where,\neg isnegation{,}^{\prime }{\vee }^{\prime }$
is inclusive or and '$\Rightarrow$' is implication, is

• A. Multiple valued
• B. Cannot be determined
• C. False
• D. True

Answer 1

The correct option is B Cannot be determined

$\begin{array}{ccc}p& q& \rightharpoondown p\to q\\ 0& 0& 0\\ 0& 1& 1\\ 1& 0& 1\\ 1& 1& 1\end{array}$
Now since $\rightharpoondown p\to q$ is given true, we reduce the truth table as follows:
$\begin{array}{ccc}p& q& \rightharpoondown p\to q\\ 0& 1& 1\\ 1& 0& 1\\ 1& 1& 1\end{array}$

In the reduced truth table we need to find the truth value of $\rightharpoondown p\vee (p\to q)\equiv p{}^{\prime }+(p\to q)$
$\equiv p{}^{\prime }+p{}^{\prime }+q\equiv p{}^{\prime }+q$

The truth value of p${}^{\prime }$ + q in the reduced truth table is given below:
$\begin{array}{ccc}p& q& p{}^{\prime }+q\\ 0& 1& 1\\ 1& 0& 0\\ 1& 1& 1\end{array}$

Since in the reduced truth table also, the given expression is sometimes true and somethimes false, therefore the truth value of proposition
$\rightharpoondown p\vee (p\to q)$
can not be determined.