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MATH 114 Finite Mathematics
Textbook: “Finite Mathematics: An Applied Approach” by M. Sullivan and A. Mizrahi, John Wiley & Sons, 2004 (9th edition).
1.1. Rectangular Coordinates. Lines.
Definition. The rectangular (or Cartesian) coordinate system consists of two lines, one horizontal (called the x-axis) and the other vertical (the y-axis) that intersect at the origin O.
Definition. The plane formed by the axes is called the xy-plane. Definition. Any point P in the xy-plane is specified by the coordinates (x, y). Example. Plot points (2, −1), (3, 2), etc.
Definition. A linear equation is the relation between x and y of the form Ax + By = C for some constants A, B, and C such that A and B are not both zero. Definition. The graph of the linear equation is a line. Example. A = 1, B = 2, C = −3; A = 0, B = 1, C = 3; A = − 2 , B = 0 , C = 5.
Definition. The y-intercept (0, b) of a line is the point at which the graph of the line crosses the y-axis. The x-intercept (a, 0) is where the graph crosses the x-axis.
Definition. The slope m of a line is the rate of change of y with respect to x. If P = (x 1 , y 1 ) and Q = (x 2 , y 2 ) are two distinct points and x 1 ̸= x 2 , then m = (^) xy^22 −−yx^11. Example. 45 degree line.
Different Forms of an Equation of a Line.
Name Given Equation Point-slope form Point (x 1 , y 1 ), slope m y − y 1 = m(x − x 1 ) Slope-intercept form Slope m, intercept b y = m x + b Horizontal line y-intercept b, m = 0 y = b Vertical line x-intercept a, m=infinity x = a Line between two points Points (x 1 , y 1 ) and (x 2 , y 2 ) If x 1 = x 2 , the line is vertical x = x 1 If x 1 ̸= x 2 , then m = y x^22 −−yx^11 y − y 1 = m(x − x 1 )
Example. Find the equation of the line that (a) passes through points (1, 0) and (0, 1). (b) passes through the point (2, 1) and is horizontal. (c) passes through the point (2, 1) and is vertical.
(d) intersects the y-axis at the point (0, −2) and has the slope m = −1. (e) passes through the points (1, 1) and (1, 2).
Example. A gas company calculates a customer’s monthly gas bill as $7. plus $1.50 for every gas unit used. Write the linear equation that describes this relation.
Example. Jane got a regular job and started adding $9 per week to her savings account. At the end of 11 weeks, she has $315 in savings. Write her savings as a linear function of the number of weeks since she started the job. y = 9x + 216
Example. An electric utility computes the monthly electric bill for residen- tial customers with a linear function of the number of kilowatt hours (kWh) used. One month a customer used 1560 kWh, and the bill was $118.82. The next month the bill was $102.26 for 1330 kWh used. Find the equation re- lating kWh used and the monthly bill. m = 16230.^56 = 0. 072 , y = 0. 072 x + 6. 5
Example. The American Automobile Manufacturers Association estimated that 520,000 passenger cars were exported in 2000. If exports are expected to decrease by 15,000 passenger cars per year, find the equation for this linear trend.
Example. The sporting goods store has a sale on mopeds at $725 each. Give the revenue function.
1.3. Applications.
Example (Prediction). The U.S. Census Bureau data show that in 2001 the municipal solid waste was 211.4 millions of tons, whereas in 2003 this amount increased to 220.2 millions of tons. Write down the linear model and use it to predict the gross waste in 2005. y = 4. 4 x−8593; 229 millions of tons
Example (Break-Even Point). Cox’s Department store pays $100 each for CD players. The store’s monthly fixed costs are $1000. The store sells the CD players for $200 each. (a) Find the cost function C, the price the store pays for x CD players. C = 1000 + 100x. (b) Find the revenue function R, the profit the store makes from selling x CD players. R = 200x (c) Find the break-even point, that is, how many CD players must be sold to guarantee no loss and no profit for the store. 1000 + 100x = 200x =⇒ x = 10
2.1. Systems of Linear Equations: Substitution.
Example (Mixture Problem). A gasoline supplier has two large gasohol tanks, one containing 8% alcohol and the other containing 13% alcohol. If 2000 gallons of gasohol with 10% alcohol is needed, how many gallons should be taken from each tank to provide the proper mixture? x = 1200, y = 800
Example (Investment Problem). Emily bought two stocks, one selling for $30 per share and the other for $20 per share. She invested total of $4000. The dividend from the first stock is $2 per share and from the other stock is $1 per share. Emily expects to receive a total of $220 in dividends from the two stocks. How many shares of each stock did she buy? x = 40, y = 140
Example (Mixture Problem). A store is filling an order for 50 pounds of a mixture of walnuts and pecans. The mixture will sell for $3.90 per pound. Walnuts normally sell for $4.50 per pound, and pecans for $3 per pound. If the income from selling the mixture should be the same as that from selling the nuts separately, how many pounds of each type of nut should be used in filling the order? x = 30, y = 20
Example (Investment Problem). Mr. Green has $10,000 to invest. He will invest part of this sum into secure bonds that carry 2% interest, and the rest into risky stocks with 10% interest. Mr. Green plans to receive the interest of 5% of the invested sum. How much should he invest into bonds? x = 6, 250 , y = 3, 750
2.1. Systems of Linear Equations: Elimination.
Definition. The method of elimination consists of replacing the original system of equations by an equivalent system for which the solution is easily found.
Rules for Obtaining an Equivalent System of Equations.
- Interchanging the equations.
- Multiplying each side of an equation by the same nonzero constant.
- Replacing any equation in the system by the sum of that equation and a nonzero multiple of the other equation.
Example. { 2 x + 3y = − 1 − 4 x + 5y = 13
2 r 1 ⇔
4 x + 6y = − 2 − 4 x + 5y = 13
r 1 +r 2 ⇔
2 x + 3y = − 1 11 y = 11
2 x + 3y = − 1 y = 1
x = − 2 y = 1
Example.
2 x + 3y = 5 3 x − 5 y = − 2
6 x + 9y = 15 − 6 x + 10y = − 4
2 x + 3y = 5 19 y = − 19
2 x + 3y = 5 y = 1
x = 1 y = 1
2.2. Matrix Method.
Definition. A matrix is a rectangular array of numbers (called entries). Am m × n matrix has m rows and n columns.
Definition. A system of linear equations can be represented in a matrix form (by specifying an augmented matrix). Example.
{ 2 x + 3y = − 1 − 4 x + 5y = 13
[
]
The solution of this system is
x = − 2 y = 1
which is equivalent to the aug-
mented matrix
[
]
Definition. The matrix method of solving a system of linear equations con- sists of replacing the original augmented matrix by an equivalent matrix for which the solution of the system is easily found. The replacement is done by means of row operations:
- Interchanging any two rows.
- Replacing a row by a nonzero multiple of that row.
- Replacing a row by the sum of that row and a nonzero multiple of some other row.
Example.
[
]
[
]
[
]
[
]
[
]
[
]
Example.
[
]
[
]
[
]
[
]
[
]
[
]
Example.
x + 3y = 5 − 2 x + 6y = 8
[
]
[
]
[
]
[
]
[
]
[
]
[
a 1 a 2... ar
]
b 1 b 2
... br
=^ a^1 b^1 +^ a^2 b^2 +^ · · ·^ +^ arbr.
Definition. Multiplication of two matrices is defined if the number of columns of the first matrix equals to the number of rows of the second matrix. The product of a m × r and r × n matrices is a m × n matrix which entry in row i and column j is the product of the ith row of the first matrix and the jth column of the second one.
Example. Multiply (a)
[
]
impossible
(b)
Example. Costs, in dollars, for radio (per minute), newspaper (per col- umn inch), and TV (per minute) ads in two cities are given in the following matrix: Radio Newspaper TV City A ⌈ 30 20 120 ⌉ City B ⌊ 25 18 140 ⌋ (a) Ads are run five times in each of these media in City A and eight times in each of the media in City B. Find the total amount spent on radio, news-
paper, and TV ads.
[
]
[
]
[
]
(b) The ads are run 30, 40, and 60 times, respectively, in each of the cities in January, and 20, 12, and 22 times in February. Find the total amount spent in
each city in January and February.
[
]
[
]
2.6. The Inverse of a Matrix.
Definition. A 2 × 2 identity matrix is I =
[
]
. In general, n × n iden-
tity matrix has ones on the main diagonal and zeros everywhere else. The identity matrix has the property that AI = IA = A. Definition. The inverse of a square n×n matrix A is A−^1 such that AA−^1 = A−^1 A = I. Note. Inverses are defined only for square matrices.
Proposition.
[
a b c d
]− 1
= (^) ad−^1 bc
[
d −b −c a
]
Proof:
[
a b c d
]
1 ad−bc
[
d −b −c a
]
[
]
and (^) ad−^1 bc
[
d −b −c a
] [
a b c d
]
[
]
Example.
[
]− 1
= (1)(5)−^1 (3)(2)
[
]
[
]
Method to Find the Inverse of an n × n Matrix A.
- Write down an augmented matrix
[
A | I
]
- Perform a sequence of row operations to reduce the A portion of the ma- trix to identity matrix. Then the matrix found in the I portion is A−^1.
Example. Find
− 1
.
Solution:
Solving Systems of Linear Equations Using Matrix Inverses.
Proposition. A system of n linear equations can be written in matrix form
as A
x 1
... xn
c 1
... cn
(^) for some n × n matrix A and some constants c 1 ,... , cn.
Example.
x + y = 2 y + z = − 1 x + 2z = 3
x y z
The solution of this system is
x 1
... xn
= A−^1
c 1
... cn
Proof: AX = C ↔ A−^1 AX = A−^1 C ↔ IX = A−^1 C ↔ X = A−^1 C.
Example. Solve
- 5 x + 0. 3 y = 1. 3 25 x + 7y = 49
Solution: In matrix form this is
[
] [
x y
]
[
]
. The inverse of [
- 5 0. 3 25 7
]
is
[
]
. The solution to the system is [ x y
]
[
] [
]
[
]
Example. A theater charges $4 for children and $8 for adults. One weekend, 900 people attended the theater, and the admission receipts totaled $5840.
Example.
x + y ≤ 5 x + 2y ≤ 8 x ≥ 0 y ≥ 0
− 2 x + y ≤ 1 x + y ≤ 4 y ≥ 2
x > 1 x ≤ 4 y ≥ 0
x ≥ y x < 2 y x ≤ 3
3.2. A Geometric Approach to Linear Programming Problems.
Definition. A linear programming problem consists of maximizing (or min- imizing) an objective function z = Ax + By under some constraints given as a system of linear inequalities.
Example. An appliance store has the storeroom capacity limited to 50 items. Each washer takes 2 hours to unpack and set up, and each dryer takes 1 hour. There are 80 hours of employee time available. Washers sell for $300 each, and dryers sell for $200 each. How many of each should be ordered to maximize the revenue? x=washers, y=dryers, z = 300x+200y →max, x+ y ≤ 50 , 2 x + y ≤ 80 , x ≥ 0 , y ≥ 0.
The Fundamental Theorem of Linear Programming. The solution to a linear programming problem is located as a corner point of the feasible region.
Steps for Solving a Linear Programming Problem.
- Specify the objective function and determine whether it should be maxi- mized or minimized.
- Determine the set of constraints in the form of a system of linear inequal- ities.
- Graph the feasible regions and find the coordinates of the corner points.
- Calculate the value of the objective function in each corner point.
- Select the maximum (or minimum) value of the objective function.
In our example,
Corner Point Objective Function (0,0) 0 (0,50) 10, (30, 20) 13, (40,0) 12,
Answer: 30 washers and 20 dryers.
3.3. Applications.
Example. Jack has a casserole and salad dinner. Each serving of casse- role contains 250 calories, 2 milligrams of vitamins, and 9 grams of protein. Each serving of salad contains 30 calories, 6 milligrams of vitamins, and 1 gram of protein. Jack wants to consume at least 20 milligrams of vitamins
and 12 grams of protein but keep the calories at a minimum. How many servings of each food should he eat?
Corner Point Objective Function (0,12) 360 (1,3) 340 (10,0) 2500
Answer: 1 casserole and 3 salads.
Example. A biology department buys mice and rats for experimental pur- poses. It buys at least three times as many mice as rats. Each mouse costs $2, each rat costs $5, and the departmental budget dictates that no more than $110 be spent on such purchases. If each mouse can be used for three experiments and each rat for two, how many of each should be purchased to maximize the total number of experiments that can be run?
Corner Point Objective Function (0,0) 0 (55,0) 165 (30, 10) 110
Answer: 55 mice and no rats.
Example. A company that makes desk lamps and floor lamps has 1200 hours of labor and $4200 to purchase materials each week. It takes 0.8 hour of labor to make a desk lamp and 1 hour to make a floor lamp. The materials cost $4 for each desk lamp and $3 for each floor lamp. The company makes a profit of $2.65 on each desk lamp and $3.15 on each floor lamp. How many of each should be made each week to maximize profit?
Corner Point Objective Function (0,0) 0 (0,1200) 3780 (375, 900) 3828. (1050,0) 2782.
Answer: 375 desk lamps and 900 floor lamps.
6.1. Sets.
Definition. A collection of objects (elements) is called a set. Notation. A = {a, b, c}, a ∈ A, d ̸∈ A. Example. A = { 0 , 1 , 2 , 3 , 4 , 5 }, 3 ∈ A, 8 ̸∈ A. Definition. Two sets are equal if they have the same elements. Notation. A = B, C ̸= D. Example. A = {n | n^2 ≤ 5 , n integer}, B = {− 2 , − 1 , 0 , 1 , 2 }, A = B C = { 1 , 3 , 4 }, D = { 2 , 3 , 5 , 6 }, C ̸= D. Definition. A set A is a subset of B (A ⊆ B), if every element of A is also
16 bought both. How many people bought either a t-shirt or pants?
Application of the Venn Diagram. Example. Of 50 students surveyed, 27 owned a laptop, 39 owned a graphi- cal calculator, and 25 owned both. How many students (a) did not own a calculator? (b) owned a calculator but did not own a laptop? (c) owned neither? (d) owned one or the other but not both?
Example. The SWR Group advertised for applicants for secretarial, cler- ical, and typist positions. Respondents could apply for one or more of the positions. The responses were as follows: 12 applied for secretary, 10 for clerk, 14 for typist, 7 for secretary and typist, 5 for secretary and clerk, 3 for clerk and typist, 2 for all three, 3 for none of the positions. (a) What was to total number of respondents? (b) How many applied for the typist position only? (c) How many applied for the secretary position but not the clerk position? (d) How many applied for exactly one position? (e) How many applied for exactly two positions? (f) How many applied for at least one of the positions?
Example. In the course of a day, 40 careful bulls entered a particular china shop. Fifteen of them broke a precious vase, 18 of them broke an invaluable plate, 9 broke both a vase and a plate. All those that didn’t break a vase, broke a priceless cup, but none of them broke both a vase and a cup. How many bulls broke (a) a vase or a plate or both? (b) a plate and a cup? (c) a vase only?
6.3. The multiplication Principle.
Proposition. If a task consists of a sequence of k choices in which there are n 1 selections for the 1st choice, n 2 selections for the 2nd choice,... , nk selections for the kth choice, then the task of making these selections can be done in n 1 · n 2 ·... · nk different ways.
Draw a tree diagram to illustrate this principle.
Example. How many phone numbers can be put on 555-xxxx exchange? Example. How many four-digit PINs are possible if zero cannot be used as the first digit and no digit may be repeated? 4536 Example. Three different novels and two different manuals are to be ar- ranged on a shelf. In how many ways can it be done if a novel is to occupy the middle position? 4 · 3 · 3 · 2 · 1 = 72 Example. Suppose that a license plate is made up of 2 letters followed by 5 digits. How many different such plates can be made? Example. How many 7-digit phone numbers exit that contain at least 1 zero? 10^7 − 97 = 5, 217 , 031 Example. Two-digit numbers are to be made up from the digits 1, 4, 5, 7, and 9. (a) How many such numbers exist if repetitions are allowed? (b) if repetitions are not allowed? (c) How many even numbers are possible?
Example. A nursery rhyme starts as follows:
As I was going to St. Ives I met a man with seven wives. Each wife had seven sacks. Each sack had seven cats. Each cat had seven kittens.
How many kittens did the traveller meet?
Example. 100,000 people are to be given an ID in which one letter is fol- lowed be n digits. What is the smallest possible value of n?
6.4. Permutations.
Definition. A permutation is an ordered arrangement of r objects chosen from n objects. Notation. P (n, r). Definition. A factorial n! is defined as n! = n(n − 1)(n − 2) · · · · · 1. Example. 0! = 1, 1! = 1, 2! = 2, etc. Proposition. P (n, r) = (^) (nn−!r)!. Proof: There are n balls and r boxes. There are n choices for the 1st box, n − 1 choices for the 2nd, etc. 2
Example. In how many ways can 4 of 8 books be arranged on a shelf? 1680 Example. In how many ways can 5 people sit in a row? Example. (a) An art appreciation class is asked to rank the paintings 1 through 7. How many different rankings are possible? 5040 (b) If the students are asked to rank only the top three, how many rankings are possible? 210
Example. A police department consists of ten officers. Five officers pa- trol the streets, two work at the station, and three are on reserve. (a) How many different divisions are possible? (^) 5! 2! 3!10! = 2, 520
(b) How many different divisions are possible if officer Larson patrols the streets?
4
2
Example. A committee of four is to be selected from among eight students and a professor. In how many ways can it be done if (a) the professor cannot be on the committee?
4
(b) the professor must be on the committee?
3
(c) there are no restrictions?
4
Example. How many distinct “words” can be formed using all letters in (a) TENNESSEE? Solution: There are four groups of letters: T-1, E-4, N-2, S-2. Thus, the answer is (^) 1! 4! 2! 2!9! = 3, 780 (b) STATISTICS? S-3, T-3, A-1, I-2, C-1, (^) 3! 3! 1! 2! 1!10! = 50, 400
7.1. Sample Spaces.
Definition. A random experiment is a procedure that (1) can be repeated as many times as we want and (2) has a well-defined set of possible outcomes, but outcomes are uncertain on every trial. Example. Flipping a coin: outcomes are H or T; rolling a die: outcomes are 1, 2, 3, 4, 5, or 6. Definition. A sample space S is a set of all possible outcomes of a random experiment. Examples. (1) Flipping a coin once. S = {H, T } (2) Flipping a coin twice. S = {HH, HT, T H, T T } (3) Flipping a coin three times. S{HHH, HHT, HT H, T HH, HT T, T HT, T T H, T T T } (4) Tossing a die. S = { 1 , 2 , 3 , 4 , 5 , 6 } (5) Tossing two dice. S = {(1, 1), (1, 2),... , (1, 6),... , (6, 1), (6, 2),... , (6, 6)} (6) Measuring the lifetime of a computer. S = {x : 0 ≤ x < ∞}.
Assignment of Probabilities. Definition. Suppose the sample space S has n outcomes given by S = {e 1 , e 2 ,... , en}. To each outcome we assign a real number P(e), called the probability of the outcome e, satisfying: (1) P(e) ≥ 0 for every e ∈ S, (2) P(e 1 ) + · · · + P(en) = 1. Example. Flip a coin. S = {H, T }, P(H) = P(T ) = 1/ 2. Such a coin is called fair. Example. Flip a biased coin, e.g., P(H) = 0. 8 , P(T ) = 0.2. Example. Toss a fair die. P(1) = · · · = P(6) = 1/6. Example. Toss a loaded die, e.g., P(1) = · · · = P(5) = 0. 1 , P(6) = 0.5.
Constructing a Probability Model of a Random Experiment. Definition. A probability model of an experiment consists of the sample space and the assignment of probabilities to each outcome. Example. A fair coin is tossed. Write down the probability model. Solution: S = {H, T }; P(H) = P(T ) = 1/2. Example. A fair coin is flipped two times. Write down the probability model. Solution. S = {HH, HT, T H, T T }; P(HH) = · · · = P(T T ) = 1/ 4. Example. A fair coin is tossed until a head or three tails appear. Write down the probability model. Solution: S = {H, T H, T T H, T T T }; P(H) = 1/ 2 , P(T H) = 1/ 4 , P(T T H) = 1 / 8 , P(T T T ) = 1/ 8.
Probability Models Involving Equally Likely Outcomes.
Definition. When the same probability is assigned to every outcome of a random experiment, the outcomes are called equally likely outcomes. Example. One card is drawn at random from a deck of cards. S = { 2 ♣, 2 ♠, 2 ♡, 2 ♢,... , A♣, A♠, A♡, A♢}; P(card) = 1/ 52. Definition. An event is a subset of S. Notation. An event is denoted A, B, C, D, E, F, G. Example. Two coins are flipped. Event A is to see at least one head. List the outcomes in A. Solution: A = {HT, T H, HH}. Example. A card is drawn. An event B is to draw an ace or the queen of spades. List all the outcomes in B. Solution: B = {A♣, A♠, A♡, A♢, Q♠}.
Probability of an Event E in a Sample Space with Equally Likely Outcomes. If the sample space S has n equally likely outcomes, and the event E has m outcomes, then the probability of the event E, P(E) = m/n =number of outcomes in E/number of all outcomes= c(E)/c(S). Example. A box contains 3 red, 4 green and 6 blue marbles. One marble is drawn at random. What is the probability to choose a green marble? Solution: Let G=a green marble is picked, c(G) = 4. Each marble is equally likely to be chosen, c(S) = 13. Therefore, P (G) = 4/13. Example. A card is drawn from a deck of cards. Find the probability that (a) an ace is drawn. Solution: Each card is equally likely to be drawn, so the probability to drawn one particular card is 1/52. There are four aces. Therefore, P(ace) = 4/52 = 1 /13. (b) a heart is drawn. P(heart) = 13/52 = 1/ 4.
7.2. Probability of an Event.
sen desk has (a) at least one defect, (b) neither kind of defect.
Odds. If the odds for (or in favor of) E are a to b, then P(E) = (^) a+ab.
If the odds against E are a to b, then P(E) = (^) a+bb. Example. If the odds for rain today are estimated to be 1 to 3, what is the probability that (a) it rains today? (^) 1+3^1 = 1/ 4
(b) it doesn’t rain today? (^) 1+3^3 = 3/4. Example. A single card is dealt. Find the odds for the card being a king. P(king) = 1/13, odds are 1 to 12.
7.3. Probabilities Problems Using Counting Techniques.
Example. From a group of five women and seven men, randomly choose five people. What is the probability that (a) two women and three men are chosen?
( 5 2
3
5
(b) women are not chosen?
( 5 0
5
5
(c) Mr. N. is chosen?
( 11 4
5
(d) Mr. N. is the only man chosen?
( 5 4
5
Example. From among three conservatives and five liberals, a committee of three is to be selected. What is the probability that at least two are liberals?
P(exactly two liberals) + P(exactly three liberals) =
(^31 )(^52 )
(^83 )
(^30 )(^53 )
(^83 )
10 56 =^
40 56 = 0.^71.
Example. Four cards are dealt. Find the probability that at most two are face cards. P(at most two face cards) = 1 −
[
P(three face cards) + P(four face cards)
]
[(^123 )(^401 )
(^523 )
(^124 )
(^524 )
]
[(220)(40)
270725 +^
495 270725
]
Example. A coin is tossed four times. What is the probability that (a) exactly two heads turn up?
P(exactly two heads) =
(^42 )
24 =^
6 16 = 0.^375. (b) at least one tail turns up? P(at least one tail) = 1 − P(no tails) = 1 − 161 = 1516 = 0. 9375.
Example. License plates are made with three letters followed by three dig- its. What is the probability that a license plate (a) starts with the letter A and ends with the digit 8?
(26)(26)(10)(10) (26)^3 (10)^3
(b) has no letter or digit repeated?
(26)(25)(24)(10)(9)(8) (26)^3 (10)^3
(c) doesn’t contain the letter Z or the digit 0?
(25)^393 (26)^3 (10)^3
Example. (Birthday Problem). Find the probability that in a group of r people at least two have the same birthday.
P = 1 −
(365)(364)... (365 − r + 1) (365)r^
If, e.g., r = 5, P = 0.027.
7.4. Conditional Probability.
Definition. A conditional probability of an event E given that an event F happened is
P(E| F ) =
P(E ∩ F )
P(F )
Example. In a group of 200 students, 40 are taking English, 50 are taking Math, and 12 are taking both. Given that a student is taking English, what is the probability that he is taking Math?
Example. Draw a marble from a box containing 3 green, 1 white, and 5 black marbles. If the drawn marble is not white, find the probability that it is green. Do the calculations in two ways:
(a) by definition of conditional probability.
