1.1, Logical Reasoning.

These are the answers to the exercises in section 1.1, Logical Reasoning

- Truth tables

1. $\lnot P\Rightarrow Q$
P |
Q |
$\lnot P$ |
$\lnot P\Rightarrow Q$ |
---|

T |
T |
F |
T |

T |
F |
F |
T |

F |
T |
T |
T |

F |
F |
T |
F |

2. $P\Rightarrow \lnot Q$
P |
Q |
$\lnot Q$ |
$P\Rightarrow \lnot Q$ |
---|

T |
T |
F |
F |

T |
F |
T |
T |

F |
T |
F |
T |

F |
F |
T |
T |

3. $(P\lor Q)\Rightarrow R$
P |
Q |
R |
$P\lor Q$ |
$(P\lor Q)\Rightarrow R$ |
---|

T |
T |
T |
T |
T |

T |
T |
F |
T |
F |

T |
F |
T |
T |
T |

T |
F |
F |
T |
F |

F |
T |
T |
T |
T |

F |
T |
F |
T |
F |

F |
F |
T |
F |
T |

F |
F |
F |
F |
T |

4. $(P\land Q)\Rightarrow (R\lor S)$
P |
Q |
R |
S |
$P\land Q$ |
$R\lor S$ |
$(P\land Q)\Rightarrow (R\lor S)$ |
---|

T |
T |
T |
T |
T |
T |
T |

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

5. $(P\Rightarrow Q)\Leftarrow (R\Rightarrow S)$
P |
Q |
R |
S |
$P\Rightarrow Q$ |
$R\Rightarrow S$ |
$(P\Rightarrow Q)\Leftarrow (R\Rightarrow S)$ |
---|

T |
T |
T |
T |
T |
T |
T |

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

2. Negated statements

- $\lnot P\lor \lnot Q$
- $(\lnot P\land \lnot Q)\lor (\lnot R\lor \lnot S)$
- $(\lnot P\lor Q)\land (R\land \lnot S)$