### Your Turn

2.1

2.2

2.3

2.6

2.10

1.

If our friends did not come over to watch the game, then my roommates ordered pizza or I ordered wings.

2.11

3.

((\text{~}p) \vee (\text{~}q)) \leftrightarrow (\text{~}(p \wedge q)); this is another example of De Morganâ€™s Laws and it is always true.

2.12

2.15

2.16

3.

p | q | r | \text{~}q | p\, \wedge \text{~}q | (p \wedge \text{~}q) \vee r |
---|---|---|---|---|---|

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

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

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

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

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

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

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

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

2.17

2.18

2.19

1.

Valid

p | q | \text{~}{p} | \text{~}{p} \vee {q} | {q} \to \left( {\text{~}{p} \vee {q}} \right) |
---|---|---|---|---|

T | T | F | T | T |

T | F | F | F | T |

F | T | T | T | T |

F | F | T | T | T |

2.

Not valid

p | q | \text{~}{p} | {q} \wedge {p} | \text{~}{p} \to \left( {{q} \wedge {p}} \right) |
---|---|---|---|---|

T | T | F | T | T |

T | F | F | F | T |

F | T | T | F | F |

F | F | T | F | F |

2.20

2.21

1.

Valid

p | q | {p} \wedge {q} | \text{~}\left( {{p} \wedge {q}} \right) | \text{~}{p} | \text{~}{q} | \text{~}{p} \vee \text{~}{q} | \text{~}\left( {{p} \wedge {q}} \right) \leftrightarrow \left( {\text{~}{p} \vee \text{~}{q}} \right) |
---|---|---|---|---|---|---|---|

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

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

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

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

2.

Not Valid

p | q | \text{~}{p} | {q} \wedge {p} | \text{~}{p} \leftrightarrow \left( {{q} \wedge {p}} \right) |
---|---|---|---|---|

T | T | F | T | F |

T | F | F | F | T |

F | T | T | F | F |

F | F | T | F | F |

3.

Valid

p | q | {p} \to {q} | \text{~}{p} | \text{~}{p} \vee {q} | \left( {p \to q} \right) \leftrightarrow \left( {\text{~}p \vee q} \right) |
---|---|---|---|---|---|

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

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

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

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

4.

Valid

p | q | r | \text{~}{p} | \text{~}{q} | {p} \wedge {q} | \left( {{p} \wedge {q}} \right) \to {r} | \text{~}{p} \vee \text{~}{q} | \left( {{\text{~}p} \vee {\text{~}q}} \right) \vee {r} | \left( {{p} \wedge {q} \to {r}} \right) \leftrightarrow \left( {{\text{~}p} \vee {\text{~}q} \vee {r}} \right) |
---|---|---|---|---|---|---|---|---|---|

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

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

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

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

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

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

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

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

2.22

1.

p \to q is logically equivalent to \text{~}q \to\text{~}p.

p | {q} | {p} \to {q} | \text{~}{q} | \text{~}{p} | \text{~}{q} \to \text{~}{p} | \left( {p} \to {q} \right) \leftrightarrow \left( {\text{~}{q} \to \text{~}{p}} \right) |
---|---|---|---|---|---|---|

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

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

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

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

2.

p \to q is not logically equivalent to p \vee \text{~}q.

p | q | {p} \to {q} | \text{~}{q} | {p}\rm \vee \text{~}{q} | \left( {{p} \to {q}} \right) \leftrightarrow \left( {{p}{\rm{ }} \vee \text{~}{q}} \right) |
---|---|---|---|---|---|

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

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

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

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

2.23

2.25

2.26

2.27

2.28

2.29

1.

Eric needs to replace the light bulb, and Marcos did not leave the light bulb on all night, and Dan did not break the light bulb.

2.30

1.

p | q | {p} \vee {q} | \text{~}\left( {{p} \vee {q}} \right) | \text{~}p | \text{~}q | \text{~}p \wedge \text{~}q | \text{~}({p} \vee {q}) \leftrightarrow \left( {\text{~}p \wedge \text{~}q} \right) |
---|---|---|---|---|---|---|---|

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

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

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

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

2.31

2.32

2.33