Question

If p, q, r have truth values T, F, T respectively then which of the following is true?
(a) $ \left( p\to q \right)\wedge r $
(b) $ \left( p\to q \right)\wedge \sim r $
(c) $ \left( p\wedge q \right)\wedge \left( p\vee r \right) $
(d) $ q\to \left( p\wedge r \right) $

Answer 1

Hint: To find the correct option, we have to put the truth values of p, q, r in each option and then see which option will give the final an answer as true. Some properties of $ ''p\wedge q'' $ : when any of “p or q” becomes false then the overall statement will be false. And when both “p and q” are true then this operation will be true. The property of the following operation $ \left( p\to q \right) $ is as follows: this operation is always true except when “q” is false and “p” is true. The truth value of operation $ \left( p\vee q \right) $ is that this operation is false when both “p and q” are false otherwise this statement is always true.

Complete step by step solution:
In the above problem, it is given that: p, q, r have truth values T, F, T respectively and we have asked to find the option which is true. So, we are going to check the options one by one and then come to a solution.
Checking option (a) we get,
(a) $ \left( p\to q \right)\wedge r $
Substituting the given truth values of p, q and r in the above operation we get,
 $ \left( T\to F \right)\wedge T $
Now, simplifying the above operation, by firstly simplifying $ \left( p\to q \right) $ . We know that $ \left( p\to q \right) $ is false when “p is true” and “q is false” and in the above it is given that “p is true” and “q is false” so $ \left( p\to q \right) $ is false. Now, we are going to use this result in simplifying this option.
 $ F\wedge T $
We know that, in $ ''p\wedge q'' $ , if any of p or q is false then the whole statement becomes false so the above statement $ F\wedge T $ is false.
Hence, the truth value of option (a) is false.
As we are looking for the true option so now, we are moving towards the next option.
(b) $ \left( p\to q \right)\wedge \sim r $
First statement $ \left( p\to q \right) $ in the above option we have already solved in the previous option and whose answer is false. So, substituting the simplification of $ \left( p\to q \right) $ in the above option we get,
 $ F\wedge \sim r $
It is given that the truth value of “r” is “true” and the $ ''\sim '' $ before “r” means negation of “r”. As the truth value of “r” is given as true so $ ''\sim r'' $ means the negation of truth value of “r” so result of $ ''\sim r'' $ is false. Now, substituting $ ''\sim r'' $ as false in $ F\wedge \sim r $ we get,
 $ F\wedge F $
The statement $ ''p\wedge q'' $ is false when both “p and q” are false so the above statement $ F\wedge F $ is also false.
Hence, the result of option (b) is false.
Now, checking option (c) we get,
(c) $ \left( p\wedge q \right)\wedge \left( p\vee r \right) $
First of all, we are going to simplify $ \left( p\wedge q \right) $ in which “p is true and q is false” so this statement becomes false.
Now, to simplify $ \left( p\vee r \right) $ by substituting “p as true and r as true” we get,
 $ \left( T\vee T \right) $
We know that in $ \left( p\vee r \right) $ , if “p and r both are true” then the complete statement is true. Hence, $ \left( p\vee r \right) $ is true. Now, substituting $ \left( p\wedge q \right) $ as false and $ \left( p\vee r \right) $ as true in the above option we get,
 $ \left( F \right)\wedge \left( T \right) $
We know that, if any of “p or q” in $ \left( p\wedge q \right) $ is false then the whole statement is false. Hence, this option is false.
Checking option (d) we get,
(d) $ q\to \left( p\wedge r \right) $
Simplifying $ \left( p\wedge r \right) $ by substituting “p as true and q as true” we get,
 $ \left( T\wedge T \right) $
The above statement is true because both “p and r are true”.
Now, substituting “q” as false and $ \left( p\wedge r \right) $ as true in the above option we get,
 $ F\to T $
We know that $ \left( p\to q \right) $ is true when “p is false and q is true”. Hence, the truth value of this option (d) is true.
Hence, the correct option is (d).

Note:
The possible mistake that could happen in the above problem is that you might forget to consider $ ''\sim '' $ option (b) where $ ''\sim r'' $ is given. Failing to consider $ ''\sim '' $ in option (b) will make your answer as incorrect.
Another thing, simplification of the above options will be easier if you solve the brackets first and then move further in the solution.