Fallacies The proposition [(p  q)  q]  p is not a tautology, because it is false when p is false and q is true. This type of incorrect reasoning is called the fallacy of affirming the conclusion. Ex. Is the following argument valid? “I you do every problem in this book, then you will learn discrete mathematics.” “You learned discrete mathematics.” Therefore, “you did every problem in this book.” Sol. Let p : “you did every problem in this book.” q : “You learned discrete mathematics” If p  q and q, then p

The proposition [(p  q)   p]   q is not a tautology, because it is false when p is false and q is true. This type of incorrect reasoning is called the fallacy of denying the hypothesis. Is it correct to assume that you did not learn discrete mathematics if you did not do every problem in this book It is possible that you learned discrete mathematics even if you did not do every problem in this book.

Rules of Inference for Quantified Statements

Ex. Show that the premises “Everyone in this discrete mathematics class has taken a course in computer science” and “Marla is a student in this class” imply the conclusion “Marla has taken a course in computer science”. Sol. Let D(x) : “x is in this discrete mathematics class” and let C(x) : “x has taken a course in computer science.” The premises are  x(D(x)  C(x)) and D(Marla). The conclusion is C(Marla).

Well Formed Formula A wff of propositional logic, also called propositional form, is a string consisting of propositional variables, connectives and parentheses used in proper manner. Ex. ((p  q)  (  p  q  r)  (  s  q)) is a wff A variable or the negation of a variable is called literal An elementary product is a conjunction of literals.  p  q, q  r   s, q are elementary products An elementary sum is a disjunction of literals  p  q, q  p  s, p are the elementary sums.

Disjunctive Normal Form (DNF) A formula which consists of a sum (disjunction) of elementary products (conjunction) is called a disjunctive normal form (DNF). Ex. Obtain the dnf of (p  q)  (  p  q). Sol. (p  q)  (  p  q)  (  p  q)  (  p  q) [ ∵ p  q   p  q]  (  p   p  q)  (q   p  q) [Distributive Law]  (  p  q)  (  p  q) [Commutative & Idempotent Law]

Conjunctive Normal Form (CNF) A formula, which consists of a product (conjunction) of elementary sums (disjunction), is called a conjunctive normal form (CNF). Ex. Obtain the cnf of the form (p  q)  (  p  q  r) Sol. (p  q)  (  p  q  r)  (p  (  p  q  r))  (q  (  p  q  r)) [Distributive Law]  ((p   p)  (p  q)  (p  r))  ((q   p)  (q  q)  (q  r)) [Distributive Law]  (p  q)  (p  r)  (q   p)  q  (q  r) [Idempotent & Complement Law]

Principal Disjunctive Normal Form Each variable occurs either negated or non negated, both negated and non negated forms of a variable do not occur together in the conjunction. These conjunctions are called minterms Also p  q and q  p are treated as the same. The conjunctions p  q, p   q,  p  q,  p   q are the minterms of p and q. A formula consisting of disjunctions of minterms only is known as principal disjunctive normal form. Also called sum-of-products canonical form.

Ex. Obtain principal disjunctive normal form for p   q. Sol. p   q  [p  (q   q)]  [  q  (p   p)]  (p  q)  (p   q)  (  q  p)  (  q   p)  (p  q)  (p   q)  (  p   q)

Principal Conjunctive Normal Form maxterm as a dual to minterm. For a given number of variables, the maxterm consists of disjunctions in which each variable or its negation, but not both, appears only once. A formula consisting of conjunction of maxterms only is known as principal conjunctive normal form. Also called the product of sums canonical form.

Ex. Find principal conjunctive normal form for (p  q) Sol. p  q = (p  q)  (q  p) = (  p  q)  (  q  p) Ex. Find principal conjunctive normal form for [(p  q)   p   q] Sol. [(p  q)   p   q]  [(p   p)  (q   p)]   q  (q   p)   q   (q   p)   q  (  q   p)   q   q  p   q   q  p  p   q