Question

Consider the following three statements
P: 5 is a prime number
Q: 7 is a factor of 192
R: L.C.M of 5 and 7 is 35
Then the truth value of which one of the following statements is true?
A. $\left( P\wedge Q \right)\vee \left( \sim R \right)$
B. $\left( \sim P \right)\wedge \left( \sim Q\wedge R \right)$
C. $\left( \sim P \right)\vee \left( Q\wedge R \right)$
D. $P\vee \left( \sim Q\wedge R \right)$

Answer 1

Hint: To solve this question, we should know the basic truth tables which are and or. We will denote true as T and false as F. The basic truth tables are

P	Q	$\sim P$	$P\vee Q$	$P\wedge Q$
T	T	F	T	T
T	F	F	T	F
F	T	T	T	F
F	F	T	F	F

By evaluating the truth value of P, Q, R in the question, we should verify option by option and identify the option which gives the truth value. For example, we know that 5 is a prime number. So, P is true. Likewise, we can check the other statements and evaluate the options.
Complete step by step answer:
We are given three statements which are
P: 5 is a prime number
Q: 7 is a factor of 192
R: L.C.M of 5 and 7 is 35
Let us consider P. We know that 5 is a prime number. So, P is true.
Let us consider Q. Let us divide 192 with 7, we get
$\begin{align}
  & \left. 7 \right)192\left( 27 \right. \\
 & \ \ \ \underline{14\ \ } \\
 & \ \ \ \ \ 52 \\
 & \ \ \ \underline{\ \ 49} \\
 & \ \ \ \ \ \ \ 3 \\
\end{align}$
We can see that 7 is not a factor of 192. So, Q is false.
Let us consider R. Let us calculate the L.C.M of 5 and 7.
$\begin{align}
  & 5\left| \!{\underline {\,
  5,7 \,}} \right. \\
 & 7\left| \!{\underline {\,
  1,7 \,}} \right. \\
 & \left| \!{\underline {\,
  1,1 \,}} \right. \\
\end{align}$
L.C.M is $5\times 7=35$
So, R is true.
We should know the basic truth tables which are and and or. We will denote true as T and false as F. The basic truth tables are

P	Q	$\sim P$	$P\vee Q$	$P\wedge Q$
T	T	F	T	T
T	F	F	T	F
F	T	T	T	F
F	F	T	F	F

Let us consider option-A
$\left( P\wedge Q \right)\vee \left( \sim R \right)$
Using the above table and truth values of given statements, we get
$\left( P\wedge Q \right)\vee \left( \sim R \right)\Rightarrow \left( T\wedge F \right)\vee \left( F \right)\Rightarrow F\vee F\Rightarrow F$
So, option A doesn’t give truth as a value.
Let us consider option-B
$\left( \sim P \right)\wedge \left( \sim Q\wedge R \right)\Rightarrow F\wedge \left( T\wedge T \right)\Rightarrow F\wedge T\Rightarrow F$
So, option B doesn’t give truth as a value.
Let us consider option-C
$\left( \sim P \right)\vee \left( Q\wedge R \right)\Rightarrow F\vee \left( F\wedge T \right)\Rightarrow F\vee F\Rightarrow F$
So, option-C is doesn’t give truth as a value.
Let us consider option-D
$P\vee \left( \sim Q\wedge R \right)\Rightarrow T\vee \left( T\wedge T \right)\Rightarrow T\vee T\Rightarrow T$
So, the answer is option-D
$\therefore $We get a truth value for option-D. So, the answer is option-D
Note:
In these types of questions where and or operators are there, there is no importance for braces. We can remove the braces and evaluate from left to right. This process can only be adapted to a question where and or are the only operators used. These types of questions may also have multiple options as we can get true value from various combinations.