Answer

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**Hint:**Here, the concept is solve by using truth table in the manner,

The proposition \[p\] and \[q\] denoted by \[p \wedge q\] is true when \[p\] and \[q\]are true, otherwise false

The proposition \[p\] and \[q\] denoted by \[p \vee q\] is false when \[p\] and \[q\] are false, otherwise true

The proposition\[p\], \[ \sim p\] is called negation of \[p\]

The implication \[p \to q\] is the proposition that is false when \[p\] is true and \[q\] is false and true otherwise.

The bi-conditional \[p \leftrightarrow q\] is the proposition that is true when \[p\] and \[q\] have the same truth values as false otherwise.

We use the above concept to find whether the statements are true or false.

**Complete step-by-step answer:**

Every statement is either true or false.

\[p\] | \[q\] | \[ \sim p\] | \[ \sim q\] | \[p \wedge \sim q\] | \[ \sim p \wedge q\] | \[\left( {p \wedge \sim q} \right) \wedge ( \sim p \wedge q)\] |

\[T\] | \[T\] | \[F\] | \[F\] | \[F\] | \[F\] | \[F\] |

\[T\] | \[F\] | \[F\] | \[T\] | \[T\] | \[F\] | \[F\] |

\[F\] | \[T\] | \[T\] | \[F\] | \[F\] | \[T\] | \[F\] |

\[F\] | \[F\] | \[T\] | \[T\] | \[F\] | \[F\] | \[F\] |

A compound proposition that is always false, no matter what the truth values of the propositions that occur is called a fallacy.

Hence the statement-I is true, \[(p \wedge \sim q) \wedge \left( { \sim p \wedge q} \right)\] is fallacy.

\[p\] | \[q\] | \[ \sim p\] | \[ \sim q\] | \[p \to q\] | \[ \sim q \wedge \sim p\] | \[(p \to q) \leftrightarrow \left( { \sim q \wedge \sim p} \right)\] |

\[T\] | \[T\] | \[F\] | \[F\] | \[T\] | \[T\] | \[T\] |

\[T\] | \[F\] | \[F\] | \[T\] | \[F\] | \[F\] | \[T\] |

\[F\] | \[T\] | \[T\] | \[F\] | \[T\] | \[T\] | \[T\] |

\[F\] | \[F\] | \[T\] | \[T\] | \[T\] | \[T\] | \[T\] |

A compound proposition that is always true, no matter what the truth values of the propositions that occur is called a tautology.

Hence the statement-II is true, \[(p \to q) \leftrightarrow \left( { \sim q \wedge \sim p} \right)\]is tautology.

**Hence the option (c) is correct, the statement-I is true, statement-II is true and the statement-II is not the correct explanation for statement-I.**

**Note:**The statements are either true or false. This is called the law of the excluded middle.

A truth table shows how the truth or falsity of a compound statement depends on the truth or falsity of the simple statements from which it’s constructed.

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