Questions & Answers

Question

Answers

A. If it is not raining then the weather is not humid.

B. It is raining if and only if the weather is humid.

C. It is not true that it is not raining or the weather is humid.

D. It is raining but the weather is not humid.

Answer
Verified

We have been given two statements- p: it is raining and q: weather is humid. We will first try to write these in a symbolic form.

If it is not raining then the weather is not humid = $\text{~}p \to \text{~}q$

It is raining if and only if the weather is humid = $p \leftrightarrow q$

It is not true that it is not raining or the weather is humid = $\text{~}\left( {\text{~}p \vee q} \right)$

It is raining but the weather is not humid = $p \wedge \text{~}q$

We have the symbolic representation of these statements, so we will now try to form the truth table of all these statements. The statements which give the same truth table are logically equivalent, and this will give us our answer.

A. If it is not raining then the weather is not humid.

p | q | ~p | ~q | ∼p ∨ ∼q | ~p→∼q |

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

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

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

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

B. It is raining if and only if the weather is humid.

p | q | ~p | ~q | ∼p ∨ ∼q | p↔q |

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

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

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

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

C. It is not true that it is not raining or the weather is humid.

p | q | ~p | ~q | ∼p ∨ ∼q | ∼(~p ∨ q) |

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

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

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

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

D. It is raining but the weather is not humid.

p | q | ~p | ~q | ∼p ∨ ∼q | p ∧ ∼q |

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

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

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

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

From these four truth tables, it is clearly visible that the tables formed by options C and D are logically equivalent, and hence the correct options are C and D.

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