Answer

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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,

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.

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,

p | q | \[ \sim \]p | \[ \sim \]q | p\[ \wedge \]q | p\[ \vee \]q | p\[ \to \]q | p\[ \leftrightarrow \]q |

T | T | F | F | T | T | T | T |

T | F | F | T | F | T | F | F |

F | T | T | F | F | T | T | F |

F | F | T | T | F | F | T | T |

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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