Discrete Mathematics

Applied combinatorial mathematics by Edwin Beckenbach

By Edwin Beckenbach

College of CaliforniaEngineering and actual sciences extension series.Includes bibliographies.Based at the Statewide lecture sequence on combinatorial arithmetic provided via the collage of California, collage Extension, Engineering and actual Sciences department, in 1962.

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9. Obtain disjunctive normal forms of (a) p ∧ (p → q) (b) ~(p ∨ q) ↔ (p ∧ q) 10. Obtain the principal disjunctive normal forms of (i) ~p ∨ q (ii) (p ∧ q) ∨ (~p ∧ r) ∨ (q ∧ r) (iii) p → ((p → q) ∧ ~(~q ∨ ~p)) 11. Obtain the principal conjunctive normal forms of (i) (~p → r) ∧ (q ↔ p) (ii) (q → p) ∧ (~p ∧ q) (iii) q ∧ (p ∨ ~q). 12. Show that (P → Q) ∧ (R → Q) and (P ∨ R) → Q are equivalent. , 2001, MKU) 13. Define Tautology and contradiction. Find which of the following is a tautology and which is a contradiction: (P ∧ Q) ∧ ⎤ (P ∨ Q), P ∨ ⎤ (P ∧ Q) 14.

4 Principal Conjunctive Normal Form The dual of a minterm is called a Maxterm. For a given number of variables the maxterm consists of disjunctions in which each variable or its negation, but not both, appears only once. Each of the maxterm has the truth value F for exactly one combination of the truth values of the variables. Now we define the principal conjunctive normal form. 6: If A is a given formula, then an equivalent formula B is called principle conjunctive normal form of A if B is a product of maxterms.

Example 2: (i) A = {x | x is a positive Integer greater then 100}. This is read as “the set of all x is a positive Integer less than 25”. (ii) B = {x | x is a complex number}. Note: Repetition of objects is not allowed in a set, and a set is collection of objects without ordering. (iii) We can describe a set by its characteristic function. ⎧1 if x ∈ A μ A(x) = ⎨ ⎩0 if x ∉ A (iv) In another method we describe the set by a recursive formula: Example 3: Let x0 = 2, x1 = 1 and xi +1 = xi + xi –1 ; i ≥ 1 and A = {xi : i ≥ 0} .

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