Assignment No. 3 on Unit-II Mathematical Logic
- 1. DMS Sunita M Dol Walchand Institute of Technology, Solapur Page 1 Assignment No. 3 Topics covered: • Introduction • Statements and Notation • Connectives – Negation, – Conjunction, – Disjunction, – Conditional, – Bi-conditional • Statement formulas and truth tables • Well-formed formulas • Tautologies 1. Using statements R: Mark is rich H: Mark is happy Write the following statements in symbolic form a. Mark is poor but happy. b. Mark is rich or unhappy. c. Mark is neither rich nor happy. d. Mark is poor or he is both rich and unhappy. 2. Construct the truth table for the following formulas a. ¬ (¬P ∨ ¬Q) b. ¬ (¬P ∧ ¬Q) c. P ∧ (P ∨ Q) d. P ∧ (Q ∧ P) e. (¬ P ∧ (¬ Q ∧ R)) ∨ (Q ∧ R) ∨ (P ∧ R) f. (P ∧ Q) ∨ (¬P ∧ Q) ∨ (P ∧ ¬Q) ∨ (¬P ∧ ¬Q) g. (Q ∧ (P → Q)) → P h. ¬ (P ∨ (Q ∧ R)) ↔ ((P ∨ Q) ∧ (P ∨ R)) 3. Given the truth values of P and Q as T and those of R and S as F, find the truth values of the following
- 2. DMS Sunita M Dol Walchand Institute of Technology, Solapur Page 2 a. P ∨ (P ∧ Q) b. (P ∧ (Q ∧R )) ∨ ¬ ((P ∨ Q) ∧ (R ∨ S)) c. (¬ (P ∧ Q) ∨ ¬R) ∨ (((¬P ∧ Q) ∨ ¬R) ∧ S) d. ((¬P ∧ Q) ∨ ¬R) ∨ ((Q ↔ ¬P) → (R ∨ ¬S)) e. (P ↔ R) ∧ (¬Q → S) f. (P ∨ (Q → (R ∧ ¬P))) ↔ (Q ∨ ¬S) 4. For what truth values will the following statement be true? ‘It is not the case that houses are cold or haunted and it is false that cottages are warm or houses are ugly’ 5. From the formulas given below select those which are well-formed and indicate which ones are tautologies and contradictions a. (P → (P ∨ Q)) b. ((P → (¬P)) → ¬P) c. ((¬Q ∧ P) ∧ Q) d. ((P → (Q → R)) → ((P → Q) → (P → R))) e. ((¬P → Q) → (Q → P))) f. ((P ∧ Q) ↔ P) 6. Produce the substitution instances of the following formulas for the given substitutions a. (((P → Q) → P) → P) ; substitute (P → Q) for P and ((P ∧ Q) → R) for Q b. ((P → Q) → (Q → P)) ; substitute Q for P and (P ∧ ¬P) for Q 7. Determine the formulas which are substitution instance of other formulas in the list and give the substitutions a. (P → (Q → P)) b. ((((P → Q) ∧ (R → S)) ∧ (P ∨ R)) → (Q ∨ S)) c. (Q → ((P → P) → Q)) d. (P → ((P → (Q → P)) → P)) e. ((((R → S) ∧ (Q → P)) ∧ (R ∨ Q)) → (S ∨ P))