2. Rita needs to determine all of the possible combinations for planning purpose

Question

2. Rita needs to determine all of the possible combinations for planning purposes for her food truck. Define the following sets:

     A = {Soup, Salad}

     E = {Fried Chicken, Hamburger, Crispy Tofu Bowl}

     C = {Cookie, Ice Cream, Fruit Cup}

      List the elements of A x E x C.

3. Give the first six terms of the following sequence:

       The first term is -3 the second term is -2. The rest of the terms are the product of the two preceding terms.

4. Give a Boolean expression for the expression below specified by the input/output table. Use the sum of minterm pression described in the text.

                 x y f(x,y)

                 0 0 1

                 0 1 0

                 1 0 0

                 1 1 1

5. Use the following relation on the set {5, 6, 7, 8}: R = { (5, 6), (5,8), (6,7), (7,8), (6, 6), (5, 7), (8,6)} to complete 5 and 6 below.

    Draw the arrow diagram for relation R.

6. Create a matrix for the relation R.

7. Consider the following pseudocode fragment to answer a-c:

          sum :=0

          For x = 1 to 4

                        sum :=sum+x

          End-for

          a. What is the value of the variable sum after the third iteration?

          b. How many iterations will the for-loop execute?

          c. What is the final value for the variable sum after executing the for-loop?

8. Consider the following pseudocode fragment to answer a-c:

         product := 2

         count := 7

         While (count >1)

                       product :=(product)(count)

                       count :=count-3

          End-while

          a. What is the value of product after the first iteration?

          b. How many iterations will the while-loop execute?

          c. What is the final value for product?

9. Find the first four terms of the recursively defined sequence a_k=a_(k-1)+2k, for all integers k2, where a_1=0.

10. Prove the following conjecture using Mathematical Induction: 3+9+15++(6n-3)=3n^2 for all integers n1

           Proof:

Explanation / Answer

(According to Chegg policy, only four subquestions will be answered. Please post the remaining in another question)

2. A = {Soup, Salad}

     E = {Fried Chicken, Hamburger, Crispy Tofu Bowl}

     C = {Cookie, Ice Cream, Fruit Cup}

A x E x C = {(Soup, Fried Chicken, Cookie),(Soup, Fried Chicken, Ice Cream),(Soup, Fried Chicken, Fruit Cup),(Soup, Hamburger, Cookie),(Soup, Hamburger, Ice Cream),(Soup, Hamburger, Fruit Cup),(Soup, Crispy Tofu Bowl, Cookie),(Soup, Crispy Tofu Bowl, Ice Cream),(Soup, Crispy Tofu Bowl, Fruit Cup),(Salad, Fried Chicken, Cookie),(Salad, Fried Chicken, Ice Cream),(Salad, Fried Chicken, Fruit Cup),(Salad, Hamburger, Cookie),(Salad, Hamburger, Ice Cream),(Salad, Hamburger, Fruit Cup),(Salad, Crispy Tofu Bowl, Cookie),(Salad, Crispy Tofu Bowl, Ice Cream),(Salad, Crispy Tofu Bowl, Fruit Cup)}.

3. First term is -3 and second term is -2.

Third term is -3 * -2 = 6.

Fourth term is -2 * 6 = -12.

Fifth term is 6 * -12 = -72.

Sixth term is -12 * -72 = 864.

9. ak = ak-1 + 2k

a1 = 0

a2 = a1 + 2*2 = 0 + 2*2 = 4

a3 = a2 + 2*3 = 4 + 2*3 = 10

a4 = a3 + 2*4 = 10 + 2*4 = 18

a5 = a4 + 2*5 = 18 + 2*5 = 28.

10. Base case: n = 1

3 = 3*12

Inductive hypothesis: Let the result be true for n = m.

3+9+15++(6m-3)=3m2

To prove: The result is true for n = m + 1

or 3+9+15++6(m-3)+(6(m+1)-3)=3(m+1)2

or 3+9+15++(6m-3)+(6m+3)=3(m+1)2

Proof: By inductive hypothesis

3+9+15++(6m-3) = 3m2

Adding 6m + 3 both sides

=> 3+9+15++(6m-3)+(6m+3) = 3m2 + 6m + 3

=> 3+9+15++(6m-3)+(6m+3) = 3(m2 + 2m + 1)

=> 3+9+15++(6m-3)+(6m+3) = 3(m+1)2

Thus the result is true for n = m + 1 and by induction for all n.

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