# MATHEMATICS

Use De Morgan’s laws to find the negation of the following statement.

Kwame will take a job in industry or go to graduate school.

Kwame will not take a job in industry and will not go to graduate school. | |||

Kwame will take a job in industry and will not go to graduate school. | |||

Kwame will not take a job in industry and will go to graduate school. | |||

Kwame will take a job in industry and will go to graduate school. | |||

Kwame will not take a job in industry or will not go to graduate school. |

Complete the truth table for each of these compound propositions.

**a) ***p*→(¬*q*∧*r*)

p |
q |
r |
¬q |
¬q∧r |
p→(¬q∧r) |

T |
T |
T |
|||

T |
T |
F |
|||

T |
F |
T |
|||

T |
F |
F |
|||

F |
T |
T |
|||

F |
T |
F |
|||

F |
F |
T |
|||

F |
F |
F |

**b) **¬*p*→(*q*→*r*)

p |
q |
r |
¬p |
q→r |
¬p→(q→r) |

T |
T |
T |
|||

T |
T |
F |
|||

T |
F |
T |
|||

T |
F |
F |
|||

F |
T |
T |
|||

F |
T |
F |
|||

F |
F |
T |
|||

F |
F |
F |

**c) **(*p*→*q*)∧(¬*p*→*r*)

p |
q |
r |
¬p |
p→q |
¬p→r |
(p→q)∧(¬p→r) |

T |
T |
T |
||||

T |
T |
F |
||||

T |
F |
T |
||||

T |
F |
F |
||||

F |
T |
T |
||||

F |
T |
F |
||||

F |
F |
T |
||||

F |
F |
F |

**d) **(*p*↔*q*)∧(¬*q*↔*r*)

p |
q |
r |
¬q |
p↔q |
¬q↔r |
(p↔q)∧(¬q↔r) |

T |
T |
T |
||||

T |
T |
F |
||||

T |
F |
T |
||||

T |
F |
F |
||||

F |
T |
T |
||||

F |
T |
F |
||||

F |
F |
T |
||||

F |
F |
F |

**e) **(¬*p*↔¬*q*)↔(*q*↔*r*)

p |
q |
r |
¬p |
¬q |
¬p↔¬q |
q↔r |
(¬p↔¬q)↔(q↔r) |

T |
T |
T |
|||||

T |
T |
F |
|||||

T |
F |
T |
|||||

T |
F |
F |
|||||

F |
T |
T |
|||||

F |
T |
F |
|||||

F |
F |
T |
|||||

F |
F |
F |

Show that each of these conditional statements is a tautology by completing the truth tables.

**a) **(*p*∧*q*)→*p*

p |
q |
p∧q |
(p∧q)→p |

T |
T |
||

T |
F |
||

F |
T |
||

F |
F |

**b) ***q*→(*p*∨*q*)

p |
q |
p∨q |
q→(p∨q) |

T |
T |
||

T |
F |
||

F |
T |
||

F |
F |

**c) **¬*p*→(*p*→*q*)

p |
q |
¬p |
p→q |
¬p→(p→q) |

T |
T |
|||

T |
F |
|||

F |
T |
|||

F |
F |

**d) **(*p*∧*q*)→(*p*→*q*)

p |
q |
p∧q |
p→q |
(p∧q)→(p→q) |

T |
T |
|||

T |
F |
|||

F |
T |
|||

F |
F |

**e) **¬(*p*→*q*)→*p*

p |
q |
p→q |
¬(p→q) |
¬(p→q)→p |

T |
T |
|||

T |
F |
|||

F |
T |
|||

F |
F |

**f) **¬(*p*→*q*)→¬*q*

p |
q |
p→q |
¬(p→q) |
¬q |
¬(p→q)→¬q |

T |
T |
||||

T |
F |
||||

F |
T |
||||

F |
F |