## Conditional and BiConditional Statements## Conditional StatementLet p and q are two statements then "if p then q" is a compound statement, denoted by p→ q and referred as a conditional statement, or implication. The implication p→ q is false only when p is true, and q is false; otherwise, it is always true. In this implication, p is called the hypothesis (or antecedent) and q is called the conclusion (or consequent).
- If a = b and b = c, then a = c.
- If I get money, then I will purchase a computer.
## Variations in Conditional Statement
As, the values in both cases are same, hence both propositions are equivalent.
As, the values of p →q in a table is not equal to q→p and ~p→~q as in fig. So both of them are not equal to p →q, but they are themselves logically equivalent. ## BiConditional StatementIf p and q are two statements then "p if and only if q" is a compound statement, denoted as p ↔ q and referred as a biconditional statement or an equivalence. The equivalence p ↔ q is true only when both p and q are true or when both p and q are false.
Since, the truth tables are the same, hence they are logically equivalent. Hence Proved. ## Principle of DualityTwo formulas A ## Note1: The two connectives ∧ and ∨ are called dual of each other. |

Idempotent laws | (i) p ∨ p≅p | (ii) p ∧ p≅p |

Associative laws | (i) (p ∨ q) ∨ r ≅ p∨ (q ∨ r) | (ii) (p ∧ q) ∧ r ≅ p ∧ (q ∧ r) |

Commutative laws | (i) p ∨ q ≅ q ∨ p | (ii) p ∧ q ≅ q ∧ p |

Distributive laws | (i) p ∨ (q ∧ r) ≅ (p ∨ q) ∧ (p ∨ r) | (ii) p ∧ (q ∨ r) ≅ (p ∧ q) ∨ (p ∧ r) |

Identity laws | (i)p ∨ F ≅ p (iv) p ∧ F≅F | (ii) p ∧ T≅ p (iii) p ∨ T ≅ T |

Involution laws | (i) ¬¬p ≅ p | |

Complement laws | (i) p ∨ ¬p ≅ T | (ii) p ∧ ¬p ≅ T |

DeMorgan's laws: | (i) ¬(p ∨ q) ≅ ¬p ∧ ¬q | (ii) ¬(p ∧ q) ≅¬p ∨ ¬q |

**Example:** Consider the following propositions

Are they equivalent?

**Solution:** Construct the truth table for both

p | q | ~p | ~q | ~p∨∼q | p∧q | ~(p∧q) |

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

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

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

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

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