What is the converse of the following statement if triangle ABC is equilateral then it is isosceles

TEAM 5 

   1.  LET P,Q,R ,S DENOTE THE FOLLOWING STATEMENTS ABOUT A PARTICULAR TRIANGLE ABC:

p:Triangle ABC is isosceles

q: Triangle ABC is equilateral

r: Triangle ABC is equiangular

TRANSLATE EACH OF THE FOLLOWING INTO AN ENGLISH STATEMENTS.

<a> q->p

Ans: If triangle ABC is equilateral then it is isosceles

<b> ~p->~q

Ans: If triangle ABC is not isosceles then it is not equilateral

<c> q<->r

Ans: The triangle ABC is equilateral if and only if it is equiangular

<d> p^~q

Ans: The triangle ABC is isosceles and it is not equilateral.

<e> r->p

Ans: If the triangle ABC is equiangular then it is isosceles.

1.  

           NEGATE AND EXPRESS EACH OF THE FOLLOWING STATEMENTS IN SMOOTH ENGLISH:

<a> Kelsey will get a good education if she puts her studies before her interest in cheerleading.

Ans: Kelsey placed her studies before her interest in cheerleading but she did not get a good education.

<b> Norma is doing her homework, and Karen is practising her piano lessons.

Ans: ~(p ^q)

        ~p v ~q

 Norma is not doing her homework or Karen is not practising her piano lessons.

<c> If Harold passes his C++ course and finishes his data structures project, then he will graduate at the end of the semester.

Ans:  p:Horald passes his C++ course

         q: He finishes his data structures project

         r: He will graduate at the and of the semester

         ~ [ ( p ^ q ) -> r ]

        P ^ q ^ ~r

      Horald did pass his C++ course and he did finish his data structures project but he did not graduate at the end of the semester.

2.    CONSIDER EACH OF THE ARGUMENTS. IF THE ARGUMENT IS VALID, IDENTIFY THE RULE OF INFERENCE THAT ESTABLISHES ITS VALIDITY. IF NOT, INDICATE WHETHER THE ERROR IS DUE TO AN ATTEMPT TO ARGUE BY THE CONVERSE OR BY THE INVERSE.

<a> Andrea can program in C++, and can program in Java

       Therefore Andrea can program in C++.

Ans: p: Andrea can program in C++

        q: she can program in Java

     STEPS:                                                                                        REASONS:

    <1> p ^ q                                                                                   premise

            <2> p                                                                                         step <1> and rule of conjunctive simplification

      <b> A sufficient condition for bubbles to win the golf tournament is that her opponent Meg not sink a birdie on the last hole.

           Bubbles won the golf tournament

          Therefore Bubble’s opponent Meg did not sink a birdie on the last hole

   Ans: p:Bubble’s to win the golf tournament

            q: opponent meg did not sink a birdie on the last hole

         STEPS:                                                                                                REASONS:

        <1> p->q                                                                                              premise

         <2> p                                                                                                  premise

         <3> q                                                                                                  not valid argument by any laws

    The given argument is invalid….

   Attempt to argue by converse.

      <c> If Ron’s computer program is correct, then he’ll be able to complete his computer science assignment in at most two hours.

            It takes Ron over two hours to complete his computer science assignment.

           Therefore Ron’s computer program is not correct.

       Ans: p: Ron’s computer program is correct

               q: He will be able to complete his assignment in at most 2 hours.

               STEPS:                                                                                                           REASONS:

         <1> p -> q                                                                                                         premise

         <2> ~q                                                                                                               premise

          <3> ~p                                                                                                               step <1> and step <2> and modus tollens

   <d> Eileen’s car keys are in his purse, or they are on the kitchen table.

          Elieen’s car keys are not on the kitchen table.

          Therefore Eileen’s car keys are in her purse.

   Ans: p: Eileen’s car keys are in her purse

           q: They are on the kitchen table

          STEPS:                                                                                                                     REASONS:

      <1> p v q                                                                                                                     premise

      <2> ~q                                                                                                                          premise

      <3> q v p                                                                                                                     step <1> and commutative law

      <4> p                                                                                                                            step <3> and step <2> and rule of disjunctive syllogism

 <e> If the interest rate falls, then the stock market will rise.

        Interest rates are not falling.

        Therefore the stock market will not rise.

 Ans: p: Interest rates fall

         q: stock market will rise

         STEPS:                                                                                                                                REASONS:

       <1> p->q                                                                                                                               premise

       <2> ~p                                                                                                                                  premise

       <3> ~q                                                                                                                                invalid statement

      The argument is invalid…attempt to argue by the inverse

3.    DETERMINE ALL THE ELEMENTS IN EACH OF THE FOLLOWING SETS:

<a> {1+(-1)^n | n £ N }

Ans: 1 + (-1)^1 = 1-1=0

        1+ (-1)^2 = 1+1=2

        1+ (-1)^3 =1-1=0

             A= {0, 2, 0, 2, 0, 2………………..}

             A = { 0 ,2  }

       <b> {n + (1/n) |n £{1, 2 , 3, 5, 7 }}

              B = {(1+1/1) , (2+1/2) , (3 +1/3), (5+1/5), (7+1/7)}

             B = { 2, 5/2, 10/3, 26/5, 50/7 }

      <c> (n^3 + n^2 | n £ { 0, 1, 2, 3, 4}}

             C= {(0+0) , (1+1) , (2^3 + 2^2 ), (3^3 +3^2)}

             C= { 0, 2, (8+4), (27+9), (64+16)}

              C= { 0, 2, 12, 36, 80}

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