Question

If \[\left( {p \wedge \sim q} \right) \wedge \left( {p \wedge r} \right) \to \sim p \vee q\] is false then the truth values of p, q and r are respectively.
A. F, T, F
B. T, F,T
C. T, T, T
D. F, F, F

Answer 1

Hint: Here we will find the answer simply by tabulating the values of the variables and then we will check for our condition. Tabulation will help us in getting all possible conditions on the same page.

Step by step solution:
Let’s tabulate the results along with the values of the variables or logic inputs we can say.

					I	II	I→II
p	q	r	\[p \wedge \sim q\]	\[p \wedge r\]	\[\left( {p \wedge \sim q} \right) \wedge \left( {p \wedge r} \right)\]	\[ \sim p \vee q\]	\[\left( {p \wedge \sim q} \right) \wedge \left( {p \wedge r} \right) \to \sim p \vee q\]
T	T	T	F	T	F	T	T
T	T	F	F	F	F	T	T
T	F	T	T	T	T	F	F
T	F	F	T	F	F	F	T
F	T	T	F	F	F	T	T
F	T	F	F	F	F	T	T
F	F	T	F	F	F	T	T
F	F	F	F	F	F	T	T

On observing the table above we conclude that the truth values of p, q and r are T, F and T respectively.

Thus option B is the correct answer.

Note:
This is the best way to solve the problem but if you are running out of time and the options are given to you then just go for options given to you directly just to save your time I would say.

p	q	r	\[p \wedge \sim q\]	\[p \wedge r\]	\[\left( {p \wedge \sim q} \right) \wedge \left( {p \wedge r} \right)\]	\[ \sim p \vee q\]	\[\left( {p \wedge \sim q} \right) \wedge \left( {p \wedge r} \right) \to \sim p \vee q\]
F	T	F	F	F	F	T	T
T	F	T	T	T	T	F	F

We can check for remaining options in case only if we don’t get the required answer. The solution we have written in mainstream is useful if you are asked more than one question on that one single expression. Otherwise use the method we have mentioned in NOTE.