The truth values of $p,q{\text{ and }}r$ for which $\left( {p \wedge q} \right) \vee \left( { \sim r} \right)$ has truth values $F$ are respectively A.$F,T,F$ B.$F,F,F$ C.$T,T,T$ D.$T,F,F$ E.$F,F,T$
Answer
Verified
Hint- This question is solved by making truth table of $p,{\text{ }}q,{\text{ }}r,{\text{ }} \sim r,{\text{ }}\left( {p \wedge q} \right){\text{ and }}\left( {p \wedge q} \right) \vee \left( { \sim r} \right)$.
Now given that, $\left( {p \wedge q} \right) \vee \left( { \sim r} \right)$ has truth values $F$ Now we have to find the truth values of $p,q{\text{ and }}r$ We will find that by using truth table, Now the possible combinations are ${2^3} = 8$ (Because variables are three.) Also we know that, $ \sim {\text{ means }}NOT \\ \wedge {\text{ means }}AND \\ \vee {\text{ means }}OR \\ $
$ p{\text{ }}q{\text{ }}r{\text{ }} \sim r{\text{ }}\left( {p \wedge q} \right){\text{ }}\left( {p \wedge q} \right) \vee \left( { \sim r} \right) \\ T{\text{ }}T{\text{ }}T{\text{ }}F{\text{ }}T{\text{ }}T \\ F{\text{ }}T{\text{ }}T{\text{ }}F{\text{ }}F{\text{ }}F \\ T{\text{ }}F{\text{ }}T{\text{ }}F{\text{ }}F{\text{ }}F \\ F{\text{ }}F{\text{ }}T{\text{ }}F{\text{ }}F{\text{ }}F \\ T{\text{ }}T{\text{ }}F{\text{ }}T{\text{ }}T{\text{ }}T \\ F{\text{ }}T{\text{ }}F{\text{ }}T{\text{ }}F{\text{ }}T \\ T{\text{ }}F{\text{ }}F{\text{ }}T{\text{ }}F{\text{ }}T \\ F{\text{ }}F{\text{ }}F{\text{ }}T{\text{ }}F{\text{ }}T \\ $ Now it is given that $\left( {p \wedge q} \right) \vee \left( { \sim r} \right)$ has truth values $F$ and we have to find the values of $p,q{\text{ and }}r$ when $\left( {p \wedge q} \right) \vee \left( { \sim r} \right)$ has truth values $F$. Now, from the truth table, we will see the respective values of $p,q{\text{ and }}r$ for $\left( {p \wedge q} \right) \vee \left( { \sim r} \right)$ has truth values $F$. Therefore, the values of $p,q{\text{ and }}r$ are $F,F,T$ respectively. Thus, the correct option is (E).
Note- Whenever we face such types of questions the key concept is that we should solve it by using a truth table. In this question we find the values of $p,q{\text{ and }}r$ by making the truth table and then we see the respective values of $p,q{\text{ and }}r$ for $\left( {p \wedge q} \right) \vee \left( { \sim r} \right)$ has truth values $F$.
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