## Mathematical Reasoning

# Implications

1. The meaning of implications "If","Only if","if and only if". A sentence with if p, then q can be written in the following ways.

(i) p implies q (denoted by p ⇒ q)

(ii) p is a sufficient condition for q

(iii) q is a necessary condition for p

(iv) ∼q implies ∼p

2. The contrapositive of a statement p ⇒ q is a statement ∼q ⇒ ∼p. The converse of a statement p ⇒ q is the statement q ⇒ p. p ⇒ q together with its converse, gives p if and only if q.

### Implications: Part 1

### Implications: Part 2

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1.** Truth table**

p | q | p ⇒ q |

T | T | T |

T | F | F |

F | T | T |

F | F | T |

2.

p | q | p ⇔ q |

T | T | T |

T | F | F |

F | T | F |

F | F | T |

- Law of converse : p ⇒ q converse q ⇒ p
- Law of Inverse : p ⇒ q Inverse ∼p ⇒ ∼q
- Law of Contrapositive : p ⇒ q Contrapositive ∼q ⇒ ∼p