Question
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If truth values of p be F and q be T. Then, truth value of \[ \sim ( \sim p \vee q)\] is equal to
A. T
B. F
C. Either T of F
D. Neither T nor F

Answer Verified Verified
Hint: Let us draw the truth table for different values of p and q. So, we can find the value of a given equation easily.

Complete step-by-step answer:
Now according to the preference order of mathematical reasoning.
Priority goes from left to right in a statement.
And the preference order is
Brackets ( )
Negation \[ \sim \]
And \[ \wedge \]
Or \[ \vee \]
Conditional \[ \to \]
Biconditional \[ \leftrightarrow \]
Now comes the question how to apply these operations. For this we are given with the truth table which states that if p and q are two given conditions and T stands for true and F stands for false then truth table for various values of p and q will be,

pq\[ \sim \]p\[ \sim \]qp\[ \wedge \]qp\[ \vee \]qp\[ \to \]qp\[ \leftrightarrow \]q
TTFFTTTT
TFFTFTFF
FTTFFTTF
FFTTFFTT


Now coming to the problem first we had to solve brackets.
And according to the preference order first we will find the value \[ \sim p\] in the bracket and after that find the value of \[ \sim p \vee q\].
So, using truth table if p is F then \[ \sim p\] will be T
Now we had to find or (\[ \vee \]) of \[ \sim p\] and q. And it is given in the question that q is T
Using truth table,
Or (\[ \vee \]) of T and T is T.
So, \[( \sim p \vee q)\] will be T.
Now finding negation of \[( \sim p \vee q)\].
As we know that the value of \[( \sim p \vee q)\] is T. So, according to the truth table negation of T will be F.
Hence, the value of \[ \sim ( \sim p \vee q)\] will be F.
So, the correct option will be B.

Note: Whenever we came up with this type of problem then to find the value of the given statement or equation efficiently first, we had to make a truth table for the given values and then use the preference order defined in mathematical reasoning to solve the equation step by step.
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