# Write each of the following three statements in the symbolic form and determine which pairs are logically equivalent: a) If it walks like a duck and it talks like a duck, then it is a duck. b) Either it does not walk like a duck or it does not talk like a duck, or it is a duck. c) If it does not walk like a duck and it does not talk like a duck, then it is not a duck.

## Solution:

Let:

p = it walks like a duck

q = it talks like a duck

r = it is a duck

Then given statements can be represented as:

a. (p ^ q) → r

b. ~p V ~q V r

c. (~p ^ ~q) → ~r

The truth tables for each of the above statements are as below:

a. (p ^ q) → r

p | q | r | p ^ q | (p ^ q) → r |
---|---|---|---|---|

T | T | T | T | T |

T | T | F | T | F |

T | F | T | F | T |

T | F | F | F | T |

F | T | T | F | T |

F | T | F | F | T |

F | F | T | F | T |

F | F | F | F | T |

b. ~p V ~q V r

p | q | r | ~p | ~q | ~p V ~q | ~p V ~q V r |
---|---|---|---|---|---|---|

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

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

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

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

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

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

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

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

c. (~p ^ ~q) → ~r

p | q | r | ~p | ~q | ~r | ~p ^ ~q | (~p ^ ~q) → ~r |
---|---|---|---|---|---|---|---|

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

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

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

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

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

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

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

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

Since the truth tables for statements a and b are same the two statements are equivalent.