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Finite Mathematics and Its Applications 12th edition Larry J. Goldstein, David I. Schneider, Martha J. Siegel, Steven Hair - Solutions

1. Suppose you borrow $2710 at 5% interest compounded quarterly. How much money should you pay back each quarter in order to pay back the loan in 3 years? 2. Loan Suppose you borrow $49,000 at 6.8% interest compounded quarterly. How much money should you pay back each quarter in order to pay back
1. Use a difference equation to derive the formula for the present value of a decreasing annuity in which R dollars is withdrawn from a savings account at the end of each interest period and where the interest rate per period is i. 2. A parachutist opens her parachute after reaching a speed of 100
A small city with current population 50,000 is experiencing a departure of 600 people each year. Assume that each year the increase in population due to natural causes is 1% of the population at the start of that year. (a) Find the difference equation for yn, the population after n years. (b) Solve
The atmospheric pressure at sea level is 14.7 pounds per square inch. Suppose that, at any elevation, an increase of 1 mile results in a decrease of 20% of the atmospheric pressure at that elevation. (a) Find the difference equation for yn, the atmospheric pressure at elevation n miles. (b) Solve
A sociological study was made to examine the process by which doctors decide to adopt a new drug. Certain doctors who had little interaction with other physicians were called isolated. Out of 100 isolated doctors, the number who adopted the new drug at the end of each month was 8% of those who had
A cell is put into a fluid containing an 8 milligram/liter concentration of a solute. (This concentration stays constant throughout.) Initially, the concentration of the solute in the cell is 3 milligrams/liter. The solute passes through the cell membrane at such a rate that each minute the
Psychologists have found that, in certain learning situations in which there is a maximum amount that can be learned, the additional amount learned each minute is 30% of the amount yet to be learned at the beginning of that minute. Let 12 units of information be the maximum amount that can be
Consider two genes A and a in a population, where A is a dominant gene and a is a recessive gene controlling the same genetic trait. (That is, A and a belong to the same locus.) Suppose that, initially, 80% of the genes are A and 20% are a. Suppose that, in each generation, .003% of gene A mutate
When a cold object is placed in a warm room, each minute its increase in temperature is 20% of the difference between the room temperature and the temperature of the object at the beginning of the minute. Suppose that the room temperature is 70oF and the initial temperature of the object is 40o
Suppose that the annual amount of electricity used in the United States will increase at a rate of 1.8% each year and that, this year, 3.2 trillion kilowatt-hours are being used. (a) Find the difference equation for yn, the number of trillion kilowatt-hours to be used during the year that is n
Suppose a coin has probability p of landing on heads. What is the probability of obtaining heads an even number of times in n tosses of the coin? Let yn be the probability of obtaining heads an even number of times in n tosses. Then y0 = 1 since, if you make no tosses, you get zero heads, and zero
1. A $204,700 mortgage with interest rate 4.8% compounded monthly is repaid with monthly payments of $1073.99. 2. A $143,000 mortgage with interest rate 4.5% compounded monthly with monthly payments of $724.56. 3. $25,000 is deposited into a savings account paying 3.9% interest compounded weekly
The popular sleep-aid drug Zaleplon has a half-life of one hour. That is, the amount of the drug present in the body is halved each hour. Suppose 10 mg of Zaleplon is taken at bedtime. (a) Find the difference equation for yn, the amount of the drug present in the body after n hours. (b) Solve the
From 1984 to 2014, the value of Berkshire Hathaway stock grew at a rate of about 18% compounded annually. Give the difference equation for an investment of $1000 earning 18% compounded annually, and determine the number of years required for the investment to more than triple?
A loan of $5614 at 6% interest compounded monthly is paid off with monthly payments of $818.12. Give the difference equation for the monthly balances and determine the number of months until the loan is repaid.
A steel rod of temperature 600oF is immersed in a large vat of water at temperature 75oF. At the end of each minute, its decrease in temperature is 30% of the difference between the water temperature and the temperature of the rod at the beginning of the minute. (a) Give the difference equation for
The birth rate in a certain city is 3.5% per year, and the death rate is 2% per year. Also, there is a net movement of population out of the city at a steady rate of 300 people per year. The population in 2016 was 5 million. (a) Give the difference equation for yn, the population after n years. (b)
Consider the sociological study of Exercise 43. Out of a group of non-isolated doctors, let yn be the percent who adopted the new drug after n months. Then yn satisfies the difference equation yn = .0025yn-1(500 - yn-1), y0 = 3. That is, initially 3% had adopted the drug. (a) What percent had
After caffeine is absorbed into the body, 13% is eliminated from the body each hour. Assume that a person drinks an 8-oz cup of brewed coffee containing 130 mg of caffeine, and the caffeine is absorbed immediately into the body. (a) Find the difference equation for yn, the amount of caffeine in the
Refer to Exercise 56. Suppose that the person drinks a cup of coffee at 7 a.m. and then drinks a cup of coffee at the end of each hour until 7 a.m. the next day. Refer to Exercise 56, After caffeine is absorbed into the body, 13% is eliminated from the body each hour. Assume that a person drinks an
1. A car loan of $9480 with interest rate 6% compounded monthly is repaid in 3 years with monthly payments of $288.40. 2. $2500 is deposited into a savings account paying 2.8% interest compounded semiannually and $1000 is added to the account at the end of each half year. 3. $1000 is deposited into
Determine which sentences are statements. 1. The number 3 is odd. 2. The 1939 World's Fair was held in Miami. 3. The price of coffee depends on the rainfall in Brazil. 4. The Nile River flows through Asia. 5. What a way to go! 6. If snow falls on the Rockies, people are skiing in Aspen. 7. Why is
Give the simple statements in each of the compound statements. 1. Lagos is the largest city in Nigeria or Alpha Centauri is the nearest star to Earth. 2. 2 + 2 = 5 and 4 < 7. Give the simple statements in each of the compound statements, and write the compound statement symbolically. 3. China is in
1. If a U.S. citizen travels overseas, then they need a valid passport. 2. The Smithsonian Museum of Natural History has displays of rocks and bugs?
Write the compound statements in symbolic form, using p and q for each of the simple statements. 1. The number 7 is odd and the number 14 is even. 2. No Amtrak trains go to Chicago or Cincinnati?
Let p denote the statement "Paris is called the City of Lights," and let q denote the statement "The Eiffel Tower is located in Paris." Write the following statements as English sentences: (a) (p (b) (p ˅ q (c) p ( q (d) p ˅ q (e) (p ˄ (q (f) ((p ˅ q)
Let p denote the statement "Ozone is opaque to ultraviolet light," and let q denote the statement "Life on Earth requires ozone." Write the following statements as English sentences: (a) p ˄ q (b) (p ˅ q (c) (p ˅ (q (d) (q ( p
1. Let p denote the statement "Papyrus is the earliest form of paper," and let q denote "The papyrus reed is found in Africa." Put the following statements into symbolic form: (a) Papyrus is not the earliest form of paper. (b) The papyrus reed is not found in Africa or papyrus is the earliest form
1. What is a logical statement? 2. Write down the truth tables of the simple logical connectives ˄, ˅, (, (, ( ?
1. Associate each of the logical connectives and, or, not, if . . . . then, and XOR with a logical symbol. 3. What is a tautology? Describe how you would prove that a statement is a tautology. 4. State De Morgan's laws for quantified statements?
1. When is p ( q a false statement? 2. What do we mean by "logical equivalence"? Explain how you might use a truth table to determine logical equivalence. 3. State De Morgan's laws. When should you use them? 4. Given the implication p ( q, what is the hypothesis? What is the conclusion? 5. Given
1. Give an example (in words) of an implication that is true, and write its contra-positive and converse, making sure that you know the difference? 2. Write the negation of p ( q without using the arrow symbol?
1. Determine which of the following are statements: (a) The universe is 1 billion years old. (b) What a beautiful morning! (c) Mathematics is an important part of our culture. (d) He is a gentleman and a scholar. (e) All poets are men. 2. Write each of the following statements in "if . . . . . ,
For the given input for A and B, determine the output for Z. LET C = 3 * A + B IF ((C > 0) AND (B > 3)) THEN LET Z = C ELSE LET Z = 100 (a) A = 4, B = 5 (b) A = 10, B = 2 (c) A = -4, B = 5 (d) A = -10, B = -2
For the given inputs, determine the output for Z. IF ((X > 0) AND (Y > 0)) OR (C ( 10) THEN LET Z = X * Y ELSE LET Z = X + Y (a) X = 5, Y = 10, C = 10 (b) X = 5, Y = -5, C = 10 (c) X = -10, Y = -5, C = 2 (d) X = 2, Y = 5, C = 4
1. Assume that the following statement is true: If the deduction is allowed, then the tax law has been revised. Give the truth value of each of the following statements if it can be determined directly from the truth value of the original: (a) If the tax law has been revised, the deduction is
Assume that the statement "All mathematicians like rap music" is true. Determine the truth value of the following from this assumption, if possible: (a) Some mathematicians like rap music. (b) Some mathematicians do not like rap music. (c) There exists no mathematician who does not like rap music?
1. Assume that the statement "Some apples are not rotten" is true. Determine the truth value of each of the following, using only that fact, if possible: (a) Some apples are rotten. (b) All apples are not rotten. (c) Not all apples are rotten. 2. Show that the argument is valid: If taxes go up,
Show that the argument is valid: I study mathematics and I study business. If I study business, then I cannot write poetry or I cannot study mathematics. Therefore, I cannot write poetry?
Show that the argument is valid: If I shop for a dress, I wear high heels. If I have a sore foot, then I do not wear high heels. I shop for a dress. Therefore, I do not have a sore foot?
Show that the argument is valid: Asters or dahlias grow in the garden. If it is spring, then asters do not grow in the garden. It is spring. Therefore, dahlias grow in the garden?
Use indirect proof to show that the argument is valid: If the professor gives a test, then Nancy studies hard. If Nancy has a date, she takes a shower. If the professor does not give a test, then Nancy does not take a shower. Nancy has a date. Therefore, Nancy studies hard?
Draw the logic circuit corresponding to the expression ((p ˅ q) ˄ (q ˄ r), and determine the output when the input values are p = 0, q = 0, r = 1?
Simplify the logic circuit in Fig. 19 by using the Table of Logical Equivalences (Table 1, Section 11.4) to derive its equivalent.Figure 19:Table 1 Rules of Inference:
1. Design a voting machine for a committee of three people. A motion is presented to the committee for a vote. Each person can vote "yes" by pushing a button (on) and may vote "no" by not pushing the button. The majority rules. The circuit that you design should light up when the motion passes.
Construct a statement equivalent to p XOR q, using only nor and AND?
Write the contra-positive and the converse of each of the statements. (a) The Yankees are playing in Yankee Stadium if they are in New York City. (b) If the Richter scale indicates that the earthquake is a 7, then the quake is considered major. (c) If a coat is made of fur, it is warm. (d) If Jane
Negate the statements. (a) If two triangles are similar, their sides are equal. (b) There exists a real number x such that x2 = 5. (c) For every positive integer n, if n is even, then n2 is even also. (d) There exists a real number x such that x2 + 4 = 0?
Determine which of the statements are tautologies? (a) p ˅ ( p (b) (p ( q) ( ((p ˅ q) (c) (p ˄ ( q) ( ( ((p ˄ q) (d) [p( (q ( r)] ( [(p ( q) ( r]
Construct a truth table for each of the following statements: (a) p ( (( q ˅ r) (b) p ˄ [q ( (r ˄ p)]
1. True or false? (a) [p ˄ ((p ˅ q)] ( q (b) [(p ( q) ˄ q] ( p 2. True or false? (a) ((q ( (p) ( (p ( q) (b) [(p ( q) ˄ (r ( p)] ( q 3. True or false? (a) [(p ( q) ˄ ( p] ( ( q (b) [(p ( q) ˄ ( q] ( ( p
Show that the expressions are statement forms according to the definition. Assume that p, q, and r are simple statements. 1. ( (p ˄ (r) ˅ q 2. (p ( (q) ˄ r 3. ((p ˅ r) ( (q ˄ r) 4. ([(p ˄ r) ( q]
Determine whether the statement forms are logically equivalent. 1. (p ˄ (q and ((p ˄ q) 2. p ˄ p and p 3. ( (p ( q) and (p ˄ q) ˅ ((p ˄ q) 4. (p ˄ q) ˅ r and p ˄ (q ˅ r)
1. How many possible truth tables can you construct for statement forms involving two variables p and q? (Consider the number of ways to complete the last column of a truth table for each of the four possible pairs of values for p and q.) 2. How many possible truth tables can you construct for
Define p | q by the following truth table:This connective is called the Sheffer stroke (or NAND). Construct truth tables for the following: (a) p | p (b) (p | p) | (q | q) (c) (p | q) | (p | q) (d) p | [(p | q) | q]
Define the connective ÆŸ by the following truth table:Construct the truth tables of the following: (a) p ÆŸ (q (b) (p ÆŸ q) ÆŸ r (c) p ÆŸ (q ÆŸ r) (d) ((p ÆŸ q) Ë„ (p ( q)
Let p denote "John Lennon was a member of the Beatles," and let q denote the statement "The Beatles came from Spain." Determine the truth values of the following: (a) p ˅ (q (b) ( p ˄ q (c) p ( q (d) ( p ( q (e) ( (p ( q) (f) (p ˅ q) ( ( q
Let m denote the statement "The Magna Carta was signed in 1995," and let p denote "The Parthenon is located in Italy." Determine the truth value of each of the following: (a) m ˄ p (b) ( m ˄ ( p (c) ( m ˄ ( p (d) ( m ( ( p
1. Suppose p ˅ q and p ˅ (q are both true. What is the truth value of p? 2. Suppose ( p ( q is true and p ˄ q is false. What is the truth value of p? 3. Suppose p ˄ r is false and q ˄ r is true. What is the truth value of p? 4. Suppose p ( ( r is true and p ˄ q is true. What is the truth
What value will be displayed when the Enter key is pressed?1.2.
Use technology to determine the value of the logical expression when (a) p is true, q is true, and (b) p is true, q is false. 1. (p ˅ q) ˅ ( p 2. (p ˄ q) ˅ ( q 3. ( q ˄ ((p ˄ q) 4. (p ˄ (p ˅ (q)
Display a truth table for the given expression? 1. p ˄ ( q 2. ( (p ˅ q) 3. p ˄ ((q ˅ r) 4. (p ˄ (r) ˅ ((q ˄ r)
Construct truth tables for the given statement forms? 1. p ˄ (q 2. ( (p ˅ (q) ˄ (p ˄ (q) 3. (p ˅ (q) ˄ q 4. (p ˄ q) ˅ (p ˄ (q)
Construct a truth table for each of the statement forms in Exercises. 1. ( p ( ( q 2. p ˅ (q ( ( r) 3. (p ( q) ( q 4. (q ( ( p
Insert parentheses in accordance with the precedence of logical operations? 1. ( p ˄ ( q ( ( p ˄ q 2. p ˄ q ( p ˅ ( q 3. ( p ˄ ( q ˅ r ( ( q ˄ r 4. p ˅ ( q ˄ r ( p ˅ r
Let p denote the true statement "Abraham Lincoln was the 16th president of the United States," and let q denote the false statement "The battle of Gettysburg took place at the O.K. Corral." Determine the truth value of each of the statement forms in Exercises. 1. ( p ( q 2. p ( q 3. q ( p 4. p ( (
Write the statement forms in symbols, using the conditional (() or the biconditional (() connective. Name the hypothesis and the conclusion in each conditional form. Let p be the statement "Sally studied" and q the statement "Sally passes." 1. Sally studied if and only if Sally passes. 2. If Sally
Let p denote the true statement "The die is fair" and let q denote the true statement "The probability of a 2 is 1/6." Write each of the statement forms in symbols, name the hypothesis and the conclusion in each, and determine whether the statement is true or false. 1. If the die is not fair, the
Give the hypothesis and the conclusion in each statement. (a) I will run a marathon only if Amy watches my dogs. (b) In order for a student to pass the course, it is sufficient that they earn a B. (c) The football team wears green uniforms if it is my favorite. (d) Isaac has wrinkled fingers
Give the hypothesis and the conclusion in each statement. (a) If I drive too fast, I receive a traffic ticket. (b) I drive too fast only if I receive a traffic ticket. (c) Having a valid license is necessary for being a delivery driver. (d) A cow has clear eyes and a wet nose if it is healthy?
State the converse of each of the following statements: (a) City Sanitation collects the garbage if the mayor calls. (b) The price of beans goes down only if there is no drought. (c) If goldfish swim in Lake Erie, then Lake Erie is fresh water. (d) If tap water is not salted, then it boils slowly?
State the converse of each of the following statements:(a) If Jane runs 20 miles, then Jane is tired.(b) Cindy loves Fred only if Fred loves Cindy.(c) Jon cashes a check if the bank is open.(d) Errors are clear only if the documentation is complete.(e) Sally's eating the vegetables is a necessary
Determine the output value of A in the program for the given input values of X and Y. LET Z = X + Y IF (Z ( 0) AND (X > 0) THEN LET A = 6 ELSE LET A = 4 (a) X = 0, Y = 0 (b) X = 8, Y = -8 (c) X = -3, Y = 3 (d) X = -3, Y = 8 (e) X = 8, Y = -3 (f) X = 3, Y = -8
For the given input values of A and B, use the program to find the output values of X. LET C = A + B IF ((A > 0) OR (B > 0)) AND (C > 0) THEN LET X = 100 ELSE LET X = -100 (a) A = 2, B = 2 (b) A = 2, B = -2 (c) A = 2, B = -5 (d) A = -2, B = -5 (e) A = -5, B = 3 (f) A = -5, B = 8
For the input values of A and B, use the program to determine the value of Y. LET C = A * B IF ((C ( 10) AND (A < 0)) OR (B < 0) THEN LET Y = 7 ELSE LET Y = 0 (a) A = -2, B = -6 (b) A = -1, B = -6 (c) A = -2, B = 6 (d) A = 6, B = -1 (e) A = 4, B = 3 (f) A = 3, B = 1
For the given input values of X and Y, use the program to determine the output value of Z. IF Y ( 0 THEN LET Z = X / Y ELSE LET Z = -1,000,000 (a) X = 6, Y = 2 (b) X = 10, Y = 5 (c) X = 5, Y = 10 (d) X = 0, Y = 10 (e) X = 10, Y = 0 (f) X = -1,000,000, Y = 1
For the given input values of A and B, use the program to determine the value of C. IF ((A < 0) AND (B < 0)) OR (B ( 6) THEN LET C = (A * B) + 4 ELSE LET C = 0 (a) A = -1, B = -2 (b) A = -2, B = 8 (c) A = -2, B = 3 (d) A = 3, B = -2 (e) A = 3, B = 8 (f) A = 3, B = -3
For the given input values for A and B, use the program to determine the output value of C. IF ((A ( 5) OR (B ( 5)) AND (B ( 10) THEN LET C = A - B ELSE LET C = -30 (a) A = 7, B = 7 (b) A = 7, B = 3 (c) A = 7, B = 12 (d) A = 4, B = 7 (e) A = 4, B = 3 (f) A = 4, B = 12
1. Show that [(p ( q) ˄ q] ( p is not a tautology. 2. Show that the distributive laws hold: (a) [p ˅ (q ˄ r)] ( [( p ˅ q) ˄ ( p ˅ r)] (b) [p ˄ (q ˅ r)] ( [( p ˄ q) ˅ ( p ˄ r)].
1. Write a statement equivalent to ( (p ˄ ( q) ( (r ˄ p), Using only ( and ˄? 2. Write an equivalent form of p ( q, using only ( and ˅?
Write each statement, using only the connectives ( and ˅?(a) (p ˄ ( q) ( p(b) (p ( r) ˄ (q ˅ r)(c) p ( [r ˄ ( p ˅ q)](d) (p ˄ q) ( ((q ˅ r)
1. Write an equivalent form of the statement without using implication (()? (p ˅ q) ( (q ˄ ( r) 2. Write an equivalent form of the statement without using implication ((). (q ( ( r) ( (p ˄ q)
1. Write an equivalent form of the statement without using implication ((). ( (p ˄ ( q) ( (p ˅ ( r) 2. Write an equivalent form of the statement without using implication ((). ( (((p ( q) ( r)
Negate the following statements: (a) Arizona borders California and Arizona borders Nevada. (b) There are tickets available or the agency can get tickets. (c) The killer's hat was white or gray?
Negate the following statements: (a) Montreal is a province in Canada and Ottawa is a province in Canada. (b) The salesman goes to the customer or the customer calls the salesman. (c) The hospital does not admit psychiatric patients or orthopedic patients?
1. Use De Morgan's laws to negate the statement p ˅ ( q ˅ r, without using parentheses. 2. Use De Morgan's laws to negate the statement (p ˅ (q) ˅ ((r), without using parentheses?
1. Negate the following statements: (a) Jeremy takes 12 credits, or Jeremy takes 15 credits this semester. (b) Sandra received a gift from Sally and a gift from Sacha. 2. Negate the following statements: (a) The plane from California to Maine was on time, and every seat on it was taken. (b) Kenneth
Negate the following statements: (a) If I have a ticket to the theater, then I spent a lot of money. (b) Basketball is played on an indoor court only if the players wear sneakers. (c) The stock market is going up implies that interest rates are going down. (d) For humans to stay healthy, it is
1. Negate the following statements: (a) Isaac Newton and Henry Ford invented the calculus. (b) James Galway or Paul Simon plays the piano. 2. For each of the following statements, give the contra positive and determine its truth value. (a) If the sum 2 + x is an odd number, then x is odd. (b) If a
For each of the following statements, give the contra positive and determine its truth value. (a) If a book does not have a list of synonyms, then it is not a thesaurus. (b) The water is salty if it comes from the Atlantic Ocean?
Give the contra-positive, the inverse, and the converse of each statement, and then give the truth value of each. (a) If a dog is small, then it is a Chihuahua. (b) If a traffic light is green, then you do not stop. (c) If we live in France, then we are French citizens?
For each statement, give the contra-positive, the inverse, the converse, and the negation. Determine the truth value in each case. (a) If a rectangle does not have four equal sides, then it is not a square. (b) If a coin is fair, then the probability of a head is 1⁄2. (c) A nation has a red and
1. During a campaign speech in Chicago on February 5, 2008, Barack Obama said the following: "Change will not come if we wait for some other person or some other time." (Quote by Barack Obama.) (a) Let p represent "Change will come," q represent "We wait for some other person," and r represent "We
Show that p ( q is logically equivalent to ( (p ˄ ( q)?
The following statements can be found in the IRS instructions for Form 1040. Give the inverse, converse, contra-positive, and negation of each. Simplify and use De Morgan's laws when possible.(a) If you do not wish to claim the premium tax credit for 2015, you do not need the information in Part
The following statements can be found in the American Medical Association's code of conduct. Give the inverse, converse, contra-positive, and negation of each. Simplify and use De Morgan's laws when possible. (a) If the patient cannot assent, physicians should explain the plan of care and tell her
Verify modus ponens (logical implication 3 in Table 2)?
Verify modus tollens (logical implication 4 in Table 2)?
Without using truth tables, show that {[(p ˅ ( q) ˄ r] ( p} ( [(p ˅ q) ˅ ( r]?
Show that (p ( q) ( [(p ˄ - q) ( c], Using the fact that c denotes a contradiction; that is, c is always false?

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