1. Discrete Mathematics Tautology and Proofs = Forhad Ahmed Khan =

2. Rule of Inference TautologyNameProof Modus ponens P=If it snows today Q=we will go skiing Solve: If it snows today, then we will go skiing (¬q ∧ (p → q)) → ¬p Modus tollens P=If it snows today Q=we will go skiing ¬Q=we will not go skiing Solve: It's not snow today ((p → q) ∧ (q → r)) → (p → r) Hypothetical syllogism P=It is raining today Q=We will not have a barbecue today R=We will have a barbecue tomorrow Solve: If it rains today, then we will have a barbecue tomorrow ((p ∨ q) ∧ ¬p) → q Disjunctive syllogism

3. Rule of Inference TautologyNameProof p → (p ∨ q) AdditionP=It is below freezing now Q=It is raining now Solve: it is either below freezing or raining now (p ∧ q) → p SimplificationP=It is below freezing now Q=It is raining now Solve: It is below freezing and raining now ((p) ∧ (q)) → (p ∧ q) ConjunctionP=If it snows today Q=we will go skiing Solve: It snows today and we will go skiing ((p ∨ q) ∧ (¬p ∨ r)) → (q ∨ r) ResolutionP=It is snowing Q=Jasmine is skiing R=Bart is playing hockey Solve: Jasmine is skiing or it is not snowing and It is snowing or Bart is playing hockey imply that Jasmine is skiing or Bart is playing hockey.

4. Rule of Inference TautologyNameProof FallaciesP=You did every problem in this book Q=You learned discrete mathematics Solve: If you do every problem in this book, then you will learn discrete mathematics