TEAM-4

                                                                   

EXERCISE-2.1

Let p,q,r,s denote the following statements

p:I finish writing my computer program  before lunch.

q:I shall play tennis in the afternoon.

r:The sun is shining.

s:The humidity is low.

Write the following in the symbolic form:

a)If the sun is shining ,I shall play tennis this afternoon.

b)Finishing the writing of my computer program before lunch is necessary for my playing tennis this afternoon.

c)Low humidity and sunshine are sufficient for me to play tennis this afternoon.

4a)r->q

4b)p->q

4c)(s^r)->q

EXCERCISE-2.2

4)For primitive statements p,q,r,s simplify the compound statement

[[[(p^q)^r]v[(p^q)^~r]]v~q]->s

  

    let (p^q)=t

 =[[[t^r]v[t^~r]]v~q]->s

 =[[t^(rv~r)]v~q]->s                  By distributive law

 =[[t^To]v~q]->s                        By inverse law

 =[t v ~q]->s                               By identity law

 =[p^qv~q]->s                           Replacing t by (p^q)

 =[p^To]->s                                By inverse law

 =p->s                                         By identity law

 =~pvs                                         Logically equivalence

EXCERCISE-2.3

For each of the following pairs of statements, use Modus Ponens or Modus Tollens to fill in the blank line so that a valid  argument is presented.

a)If  Janice has trouble starting her car, then her daughter Angela will check Janice’s spark plugs.

    Janice had trouble starting the car.

_____________________________________________

p:Janice has trouble starting her car.

q:her daughter Angela will check  Janice's spark plugs.

        STEPS                            REASONS

1)     p->q                               Premises

2)     p                                     Premises

      ___________

       therefore

3)   q                                 Steps 2 & 3 & Modus

                                               Ponens

   

   therefore

   q= her daughter Angela will check spark plugs.

4b)If Brady solved the first problem correctly ,then the answer he obtained is 137.

Brady’s answer to the first problem is not 137. 

p:Brady solved the first problem correctly.

    q:the answer he obtained is 137.

       STEPS                              REASONS

1)     p->q                               Premises

2)      ~q                                  Premises

   ___________

     therefore

3)    ~p                               Steps 1 & 2 & Modus

                                           Tollens

  ~p=Brady has not solved the first problem  correctly.

4c)If this is a repeat-until loop ,then the body of this loop is executed at least once.

_____________________________________________

Therefore the body of the loop is executed at least once.

p: this is a repeat-until loop.

  q: the body of this loop is executed at least once.

      STEPS                                 REASONS

1)   p->q                                   Premises

2)    p

   _________

     therefore

3) q                                     Modus Ponens

 so the result to be q statments because of modus ponens we must have the premises as p to apply the modus ponens rule.

   q=this is a repeat-until loop.

4d) If Tim plays basketball in the afternoon , then he will not watch television in the evening.

_____________________________________________

Therefore Tim didn’t play  basketball in the afternoon.

 p: Tim plays basketball in the afternoon.

    q: Tim will not watch television in the evening.

   ~q: Tim will watch television in the evening.

      STEPS                                 REASONS

1)   p->q                                   Premises

2)    ~q                                    Premises

   ___________

       therefore

3)    ~p                                  Modus  tollens

     ~p:Tim didn't play basketball in the afternoon.

Excercise-3.1

Which of the following statements are true?

4a)ØЄØ

     It is a true statement.

4b)ØCØ

It is a false statement.

4c)ØCØ

It is a true statement.

4d)Øε{Ø}

It is a true statement.

4e)ØC{Ø}

It is a false statement.

4f)ØC{Ø}

It is a false statement.