In the simplex method, how is a pivot column selected? A pivot row? A pivot element? Give examples of each.

**True or False**

- True or false. If all the coefficients
*a*_{1},*a*_{2}, …,*a*_{n}in the objective function*P*=*a*_{1}*x*_{1}+*a*_{2}*x*_{2}+ … +*a*_{n}*x*_{n}are nonpositive, then the only solution of the problem is*x*_{1}=*x*_{2}= … =*x*_{n}and*P*= 0. - True or false. The pivot column of a simplex tableau identifies the variable whose value is to be decreased in order to increase the value of the objective function (or at least keep it unchanged).
- True or false. The ratio associated with the pivot row tells us by how much the variable associated with the pivot column can be increased while the corresponding point still lies in the feasible set.
- True or false. At any iteration of the simplex procedure, if it is not possible to compute the ratios or the ratios are negative, then one can conclude that the linear programming problem has no solution.
- True or false. If the last row to the left of the vertical line of the final simplex tableau has a zero in a column that is not a unit column, then the linear programming problem has infinitely many solutions.
- True or false. Suppose you are given a linear programming problem satisfying the conditions:
- The objective function is to be minimized.
- All the variables involved are nonnegative, and
- Each linear constraint may be written so that the expression involving the variables is greater than or equal to a negative constant.

Then the problem can be solved using the simplex method to maximize the objective function *P* = -*C*.

- True or false. The objective function of the primal problem can attain an optimal value that is different from the optimal value attained by the dual problem.

What is the difference between the accumulated amount (future value) and the present value of an investment? Give examples of each.

**True or False**

- True or false. When simple interest is used, the accumulated amount is a linear function of time.
- True or false. Compound interest that is converted once a year is the same as simple interest.
- True or false. If interest is compounded annually, then the effective rate of interest is the same as the nominal rate of interest.
- True or false. The present value is always smaller than the future value.
- True or false. The future value of an annuity can be found by adding together all the payments that are paid into an account.
- True or false. The periodic payment
*R*where and*P*is the loan amount and*i*is the interest per period that will amortize the loan at the end of the term comprising*n*periods. - True or false. A sinking fund is the accumulated amount to be realized at some future date (the end of the term) when a fixed number of periodic payments are paid into an account earning interest at the rate of
*i*per period.

1.What is a Set? Give an example

2.When are two sets equal? Give an example of two equal sets

3.What is the empty set? Give an example

**True or False**

- True or false. A set is any collection of objects.
- True or false. A proper subset of a set is itself a subset of the set, but not vice versa.
- True or false. The empty set is a subset of every set.
- True or false. If
*A*∪*B*= ∅ , then*A*= ∅ and*B*= ∅ . - True or false. If
*A*∩*B*= ∅ , then*A*= ∅ or*B*= ∅ or both*A*and*B*are empty sets. - True or false. (
*A*∪*A*)^{c}= ∅ .^{c} - True or false. [
*A*∩ (*B*∪*C*)]= (^{c}*A*∩*B*)∩ (^{c}*A*∩*C*)^{c} - True or false.
*n*(*A*) +*n*(*B*) =*n*(*A*∪*B*) +*n*(*A*∩*B*) - True or false. If
*A ≠ B*, then*n*(*B*) =*n*(*A*) +*n*(*A*∩B).^{c} - True or false. The number of permutations of
*n*distinct objects taken all together is*n*! - True or false.
*P*(*n*,*r*) =*r*!*C*(*n*,*r*).

What is a fair game? Is the game of roulette as played in American casinos a fair game? Why?

**True or False**

- True or false. If
*E*and*F*are mutually exclusive, and*E*and*G*are mutually exclusive, then*F*and*G*are mutually exclusive. - True or false. Suppose it is determined that the probability of obtaining heads in a coin- tossing experiment using a biased coin is 0.51, then the probability of obtaining tails in a toss of the coin is 0.49.
- True or false. Suppose
*S*= {*s*_{1},*s*_{2}, …,*s*_{n}} is a finite sample space with*n*outcomes. Then 0 <*P*(*s*_{1}) +*P*(*s*_{2}) + … +*P*(*s*_{n}) < 1. - True or false.
*P*(*E*) +*P*(*E*) = 1.^{c} - True or false.
*P*(*E*∪^{c}*F*) = 1 -^{c}*P*(*E*∩*F*) - True or false. If
*A*and*B*are events in an experiment and*P*(*B*)*≠*0, then .

7. True or false.

Subject | Mathematics |

Due By (Pacific Time) | 11/12/2013 12:00 am |

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