Fsc Part 1 Mathematics (Complete Solution)

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Q3

Show that each of the following statements is a tautology

**(i) (p∧q)→p**

Solution:

p | q | p∧q | (p∧q)→p |
---|---|---|---|

T | T | T | T |

T | F | F | T |

F | T | F | T |

F | F | F | T |

The column of table shows that the statement (p∧q)→p is true for all values,so it is a tautology.

**(ii) p→(p∨q)**

Solution:

p | q | p∨q | p→(p∨q) |
---|---|---|---|

T | T | T | T |

T | F | T | T |

F | T | T | T |

F | F | F | T |

The column of table shows that the statement p→(p∨q) is true for all values,so it is a tautology.

**(iii) ∼(p→q)→p**

Solution:

p | q | p→q | ∼(p→q) | ∼(p→q)→p |
---|---|---|---|---|

T | T | T | F | T |

T | F | F | T | T |

F | T | T | F | T |

F | F | T | F | T |

The column of table shows that the statement ∼(p→q)→p is true for all values,so it is a tautology.

**(iv) ∼q∧(p→q)→∼p**

Solution:

p | q | ∼p | ∼q | p→q | ∼q∧(p→q) | ∼q∧(p→q)→∼p |
---|---|---|---|---|---|---|

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

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

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

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

The column of table shows that the statement ∼q∧(p→q)→∼p is true for all values,so it is a tautology.