TEAM 7

 Exercises2.2

7.a) if p,q are primitive statements, prove that

                   (~pvq)^(p^(p^q))  ó(p^q)

                                                                                                                               

P	q	~p	~pVq	p^q	p^(p^q)	(~pVq)^(p^(p^q)	p^q
1	1	0	1	1	1	1	1
1	0	0	0	0	0	0	0
0	1	1	1	0	0	0	0
0	0	1	1	0	0	0	0

The  above equation is logically proved

The last two coloumns are equal

                                                           OR

               Steps                                                                                     Reasons

1.           (~pVq)                                                                     Premise

2.              P                                                                            Premise

3.       Therefore:      q                                                       step1,2 Rule of                                                                                                                                                                                                               Disjunctive Syllogism

4.        p^q                                                                                 premise

5.       Therefore:  p^q                                                            step 2,3     rule of  

                                                                                                   conjunction

            

b)        write  the  dual of the logical equivalence in   (~pvq)^(p^(p^q))  ó(p^q)

       ans:   = (~p^q)v(pv(pvq))  ó(pvq)

                                                                

                                 Exercie-2.3

7.give the reasons for each step needed to show that the  following argument is valid

  [p^(p->q)^(svr)^(r->~q)] -> (svt)

Ans:

  Steps                                         reasons

1. p                                                premises

2. p->q                                           premises

3. q                                                 step1,2 Rule of detachments

4. r->~q                                          premises

5. q->~r                                   step4 &(r->~q)ó(~~q->~r)ó(q->~r)                         

6. ~r                                          step 3,5 Rule of detachment

7. svr                                         premises

8. s                                     step 6,7& rule of disjunctive syllogism

9.svt                                  step8 & rule of disjunctive Amplification

                                                           Exercise-3.1

7.Let A,B be sets from a universe U.

(a)Write a quantified statement to express the proper subset relation A C B(A subset B).

Ans: For All x[x belongs to A -> x belongs to B] ^ For Some x [x belongs to B ^ x does not belongs to A]

(b)Negate the result in part (a) to determine when A is not the proper subset of B.

Ans:For Some x[x belongs to A ^ x does not belongs to B] Or  For All x[x belongs to B -> x belongs to A]

                                               

                                                          Exercise-2.1

7.Rewrite each of the following statements as an impication in the if-then form.

a)Practicing her serve daily is a sufficient condition for Darci to have a good chance of winning the tennis tournament.

Ans:        p:Darci practices her serve daily.

                 q:she have good chance of winning the tennis                   

                   tournament.

     If Darci practices her serve daily then she will have a good chance of winning the tennis tournament.

b)Fix my air conditioner or I won't pay the rent.

Ans:   p:you do not fix my air coditioner.

            q:I shall not pay the rent.

->If you do not fix my air conditioner,then I shall not pay the rent.

c)Mary will be allowed on Larry's motorcycle only if she wears her helmet.

Ans: p: Mary is to be allowed on Larry's motorcycle.

         q: she must wear her helmet.

->If Mary is to be allowed on Larry's motorcycle,then she must wear her helmet.

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