We include a discussion of the two-phase simplex method (“big M method”) for solving linear programming problems where a fundamentally feasible starting point is not obvious. It is recommended that students perform all of these exercises; complete solutions can be found at the end of the book.
Numbers and Sets
Numbers
For each of the following numbers, which of the sets N,Z,Q,R does it belong to. For each of the following numbers, which of the sets N,Z,Q,R does it belong to.
Equations and Inequalities
In solving linear equations in one variable, our procedure is to change the given equation into a simpler equation that corresponds to the original. Even though it is a variable, we move it to the right because we want to find an expression for.
Sums
Write each of the following sums in sigma notation: i=1ci in each of the following cases:. Write each of the following sums in sigma notation: iv) Why did we require “x=1” in the previous section.
Elements of Set Theory
The set of all prime numbers smaller than 10;. v) The collection of all even prime numbers;. vi). If not, what is their intersection. i) Sis the set of all multiples of 5; T is the set of all perfect squares.
Venn Diagrams
We don't know how many people read the newspaper ads but not the flyers. Exactly five students were taking all three subjects, while 23 of them were currently not enrolled in any of the three subjects.
Averages
Also notice from the examples that neither the mean nor the median are necessarily members of the collection. Find the mean, mode, median, and standard deviation of the following data sets:.
Counting
Basic Counting Principles
If there are five questions and all questions must be answered, how many different answer sheets are possible. How many different ways can you arrange your books if all the chemistry books go to the left.
Arrangements
In how many distinct ways can you arrange the letters of the word MI SSI SSI P P I. Assuming that marbles of the same color are indistinguishable, how many different sequences of length 4 can be selected from the set. In how many ways can he arrange them if all the books on the same subject are to be grouped together.
How many ways can be arranged if the most expensive model should be in the middle.
Selections
In how many ways can he answer if he has to choose five, one of which is Question 1. In how many ways can a committee of three men and two women be chosen from six men and four women. How many ways can you decide whether you should include question 1 or question 2 (or possibly both).
How many different committees can be chosen? ii) How many of these possible committees contain exactly three Democrats and three Republicans?
More About Selections
The binomial theorem can be used to find approximate values of powers of numbers close to 1. So the number of terms containing exactly that are will be equal to the number of ways of choosing the indices1,i2,. Equation (2.6) gives us an interesting way to determine the number of subgroups of a set.
The total number of subgroups is equal to the number of subgroups with 0 elements (1, for the empty set), plus the number of subgroups with 1 element, plus the number of subgroups with 2 elements, and so on until Itself.
Probability
Events
A die is rolled and a coin is tossed, and the results are recorded. i) Write down all members of the sample space. ii) Write down all members of the event E: The die is equal. For example, two heads on the quarters and one on the nickels are recorded(2,1). i) Write down all members of the sample space. ii) Write down all the members of the events:. Otherwise, the coin is tossed a second time. i) Draw a tree diagram for this experiment. ii) List the members of the sample space. iii).
Two dice are rolled, one red and one green, and the results recorded: for example, 53 means "5 on red, 3 on green". i) Write down all terms of the sample space. ii) Write down all event members:. iii) Write all members of the events E∪F,E∩F,F∩G,F ∩G. iv) Describe the events F ∩G and F ∩G in words.
Probability Measures
What is the probability that the numbers shown total 8, given that the dice are fair. What is the probability of the following events. iii) An odd and an even number are displayed. What is the probability that the chosen card is from a month with r in the name.
What is the probability that the selected card shows a day in summer (June 22 to September 21 inclusive).
Non-uniform Probabilities
If he calls five times today, what is the probability of at least two sales. What is the probability that the researcher has a degree in mathematics, but not in computer science. ii). If six times are rolled, what is the probability that 2 is rolled exactly three times?
If you do this ten times, what is the probability that a 7 is recorded at least three times.
Counting and Probability
What is the probability that it contains a flush (all five cards of the same suit). What is the probability that two of them have the same birthday (day and month). What is the probability that two were born in the same month (but not necessarily the same year).
What is the probability that:. ii) The three R's occur together and the two T's occur together.
Stochastic Processes
What is the probability that the second card is a King, given that the first card is an Ace. What is the probability that it is a Jack, given that it is a face card (King, Queen, or Jack). What is the probability that it is a 5, given that it is not a face card (King, Queen, or Jack).
In Exercise 12, what is the probability that the light bulb chosen is defective, given that it was not chosen from box 2.
More About Conditional Probability
If he picks a screw at random, what is the probability that it is a defective screw from companyY. What is the probability that the box is round, given that the marble was blue. What is the probability that there will be neither snow nor a change in wind direction. ii).
Represent this experiment in a tree diagram and find the probability that the jelly bean is red.
Bayes’ Formula and Applications
If he arrives on time, what is the probability that he drove the caprice that morning. If the student is left-handed, what is the probability that the student is male. If she voted for the proposition, what is the probability that she is a Democrat. iii).
If she voted against the proposal, how likely is she not to be a Republican?
Further Examples of Bayes’ Formula
Behind one closed door is a valuable new car; behind each of the others is an almost worthless goat. However, after the choice is announced, but before the door is opened, the host opens one of the other two doors (not the one she chose) and reveals a goat. Third, on nights when the contestant's first choice is the door with the car, there is an equal chance that the host will open one of the other two doors.
A laboratory test for a particular disease tests positive 95% of the time when a person has the disease and tests negative 98% of the time when the person does not have the disease.
Expected Values
If the experiment is repeated n times, the expected frequency of Ei isnpi, and the expected value of the experiment is. What is the expected value of a $10 bet on a particular number. ii) Wheels at Monte Carlo usually have 37 slots, with no 00. The process continues until a non-defective switch is found. find the expected number of switches tested.
What is the expected number of girls in the family, assuming that boys and girls are equally likely.
Relations
The ratio of adjacency on the set of integers, defined by i α j if and only if. The image ofxunderf is the valuef (x); ifRis any subset ofS, thenf (R)= {f (r):r∈R}is the set of images of elements ofRand is called the image ofR. Is the given set a function fromStoS. Find the inverse of the following functions. i) f6:S→Suit Example problem 4.3, where Sis is the set of all real numbers other than 0 and 1. ii) f :R→ R+, where Ris is the set of real numbers, R+ is the set of positive real numbers and .
Is the given set a function fromStoS. In each case, the function is one-to-one.
Graphs
The term "ring graph" can be used to describe the diagram of a symmetric connection∼in which x ∼ x is sometimes true, but beware: a "ring graph" is not really a graph. This transport network can be represented as a graph whose vertices are warehouses and whose edges are roads. Competition can be represented by a graph with teams as vertices and an edge representing a game between teamvex andy.
There are two roads from AtoB, two roads from BtoC, two roads from CtoD and one road from AtoD.
Some Properties of Graphs
There are four cities, A, B, C, and D. There is one road connecting each of the following pairs: A and B, A and C, A and D, BandD, CandD. iii). In any graph or multigraph, the sum of the degrees of the vertices is equal to twice the number of edges (and consequently the sum of the degrees is an even integer). AsR is the distance between two vertices, necessarily R ≤ D. What are the degrees of the vertices in the following graphs.
What are the diameter and radius of the following graph, which is called the Petersen graph.
Euler’s Theorem and Eulerizations
IslandsBandCeach has an odd number of bridges, so one must be the start of the walk and the other the destination. It may be that we deleted every edge in the original multigraph; in this case we have already found the Euler walk. If edges still exist, there must be at least one vertex, say, that was in the original walk and is still on an edge in the new multigraph - if there were no such vertex, then there could be no connection between the edges of the walk and the edges that they remain in the new multigraph, and the original multigraph must be disconnected.
There may be more than one possible answer if it happened more than once in the first walk.
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Types of Graphs
Construction of pipelines is required so that oil can be pumped from wells to storage. The conditions of the problem imply that the graph need not contain any cycles. So the solution to the problem is to find a subgraph of the underlying graph that is a tree, and among them to find the one that is cheaper to construct.
Graph (b) is not a tree because it is not connected, although this may not be immediately obvious.
