# BIT11103+T2

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Subject Code Subject BIC 10103 / BIT 11003 Discrete Structure Item No. Tutorial 02 Date FSKTM Objective : 1) To introduce the concept Conjunctive Normal Form, CNF and Disjunctive Normal Form, DNF. 2) Understand the concept of predicate and quantifiers NAME : ……………………………………………………. MATRIC NO. : ……………………………………………………. A CNF (Conjunctive Normal Form) is a compound proposition in the form conjunctions of disjunctions of propositional variables or their negations, for example (pq) (p q)
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FSKTM Subject Code BIC 10103 / BIT 11003 Item No. Tutorial 02 Subject Discrete Structure Date Objective : 1) To introduce the concept Conjunctive Normal Form, CNF and Disjunctive Normal Form, DNF. 2) Understand the concept of predicate and quantifiers NAME : …………………………………………………….  MATRIC NO. : …………………………………………………….    A CNF (Conjunctive Normal Form) is a compound proposition in the form conjunctions of disjunctions of propositional variables or their negations, for example (p  q)  (p   q)  (  p   q). Similarly a DNF (Disjunctive Normal Form) is disjunctions of conjunctions, such as  p   q   r)  (  p   q  r)  (  p  q   r). We say that the normal form is full when no variable is missing in each bracket. Theorem:  Every compound proposition is equivalent to a CNF and to a DNF. Example:  Convert [(p  q)   p]   q to a CNF and to a DNF. Solution:  First draw the truth table. The result is p q …  [(p  q)   p]   q T T …  F T F …  T F T …  F F F …  T  A full CNF can be obtained by selecting the variables with false values from each row of the table whose result is false: (  p   q)  (p   q) and similarly a full DNF from the true: (p   q)  (  p   q). Both forms are equivalent to the given proposition: [(p  q)   p]   q  (  p   q)  (p   q)  (p   q)  (  p   q). 1. Convert each proposition to a CNF and to a DNF. a)  (p  q)  p b) (p   q)  (  p  q) c) (p  q)  r d) [(p  q)  r]  [  p  (q  r)]  2. Convert each CNF to DNF and vice versa. a) (p  q)  (  p  q) b) (p  q)  (p   q)  (  p   q) c)  p   q   r)  (  p   q  r)  (  p  q   r)  (  p   q   r) d) (p   q  r)  (  p  q  r)   3.   Let L(x,y)  bethepredicate x  likesy, andlettheuniverseofdiscoursebethesetofallpeople.Usequantifierstoexpresseachofthefollowingstatements.a)Everyonelikeseveryone.b)Everyonelikessomeone.c)Someonedoesnotlikeanyone.d)EveryonelikesGeorge.e)Thereissomeonewhomeveryonelikes.f)Thereisnoonewhomeveryonelikes.g)Everyonedoesnotlikesomeone.4.   Let S(x)  bethepredicate x  isastudent,  B(y)  thepredicate  y isabook, and H(x,y)  thepredicate x  has  y  , wheretheuniverseofdiscourseistheuniverse,thatisthesetofallobjects.Usequantifierstoexpresseachofthefollowingstatements.a)Everystudenthasabook.b)Somestudentdoesnothaveanybook.c)Somestudenthasallthebooks.d)Noteverystudenthasabook.e)Thereisabookwhicheverystudenthas.f)Alihasamathbook.5.Let B(x)  , E(x)  and G(x)  bethestatements x  isabook,  x  isexpensive, and x  isgood, respectively.Expresseachofthefollowingstatementsusingquantifiers;logicalconnectives;and B(x)  , E(x)  and G(x)  ,wheretheuniverseofdiscourseisthesetofallobjects.a)   Nobooksareexpensive.b)    Allexpensivebooksaregood.c)   Nobooksaregood.d)   Does(c)followfrom(a)and(b)?  6.Let G(x)  , F(x)  , Z(x)  ,and M(x)  bethestatements x  isagiraffe,  x  is15feetorhigher, x  isinthiszoo, and x  belongstome, respectively.Supposethattheuniverseofdiscourseisthesetofanimals.Expresseachofthefollowingstatementsusingquantifiers;logicalconnectives;and G(x)  , F(x)  , Z(x)  ,and M(x)  .a)   Noanimals,exceptgiraffes,are15feetorhigher;b)   Therearenoanimalsinthiszoothatbelongtoanyonebutme;c)   Ihavenoanimalslessthan15feethigh.d)   Therefore,allanimalsinthiszooaregiraffes.e)   Does(d)followfrom(a),(b),and(c)?Ifnot,isthereacorrectconclusion?

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