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Using truth table prove that p ˅ (q ˄ r) ≡ (p ˅ q) ˄ (p ˅ r).

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#### Solution

p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)

1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |

p |
q |
r |
q ∧ r |
p ∨ (q ∧ r) |
p ∨ q |
p ∨ r |
(p ∨ q) ∧ (p ∨ r) |

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

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

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

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

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

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

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

F | F | F | F | F | F | F | F |

The entries in the columns 5 and 8 are identical.

∴ p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)

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