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nature of mathematics

Questions and Answers of Nature Of Mathematics

What do each of the circuit symbols in Problems 4–9 mean? P on o off
What do each of the circuit symbols in Problems 4–9 mean? w
What do each of the circuit symbols in Problems 4–9 mean? .
How can symbolic logic be used to describe circuits?
Suppose I have three bags, one with two peaches, another with two plums, and a third mixed bag with one peach and one plum. Now I give the bags (in mixed-up order) to Alice, Betty, and Connie. I tell
Suppose you are given 12 coins, one of which is counterfeit and weighs a little more or less than the real coins. Using a two-pan balance scale, what is the minimum number of weighings necessary to
Daniel Kilraine was killed on a lonely road, two miles from Pontiac, at 3:30 a.m. on March 3, 1997. Otto, Curly, Slim, Mickey, and The Kid were arrested a week later in Detroit and questioned. Each
Four boys were playing marbles; their names were Gary, Harry, Iggy, and Jack. One of the boys had 9 marbles, another had 15, and each of the other two had 12.Their ages were 3, 10, 17, and 18, but
Three children were playing baseball and their names were Alice, Ben, and Cole. One of them hit a home run and broke your expensive plate glass window, and you went out to question the children.
Harry, Larry, and Moe are sitting in a circle so that they see each other. The game is played by a judge placing either a black or a white hat on each person’s head so that each person can see the
Alice: “I don’t know.”Ben: “I don’t know.”Cole: “I don’t know.”Alice: “I don’t know.”Ben: “I don’t know.”Cole: “I don’t know.”Alice: “I don’t know.”Ben: “I
Alice: “I don’t know.”Ben: “I don’t know.”Cole: “I don’t know.”Alice: “I lose.”What will Ben and Cole say next?In Problems 31–36, state what you can infer about the cards from
Alice: “I don’t know.”Ben: “I don’t win.”Cole: “I don’t win.”In Problems 31–36, state what you can infer about the cards from the given statements.
Given: AA;Prove: EUse these rules to prove the theorems in Problems 15–20. Give both statements and reasons.
Given: AE;Prove: AUse these rules to prove the theorems in Problems 15–20. Give both statements and reasons.
Given: IAEA;Prove: IUse these rules to prove the theorems in Problems 15–20. Give both statements and reasons.
Given: AI;Prove: AUse these rules to prove the theorems in Problems 15–20. Give both statements and reasons.
Given: AIII;Prove: EAUse these rules to prove the theorems in Problems 15–20. Give both statements and reasons.
Given: EA; Prove: AUse these rules to prove the theorems in Problems 15–20. Give both statements and reasons.
Three sisters visited a garden of red, white, blue, and yellow flowers. One sister observed that if any four flowers were picked, one of them would be red. Another observed that if any four were
I have a habit of getting up before the sun rises. My socks are all mixed up in the drawer, which contains 10 black and 20 blue socks. I reach into the drawer and grab some socks in the dark. How
Two doctors—an old doctor and a young doctor—were discussing an interesting case. The young doctor is the son of the older doctor, but the older doctor is not the father of the younger doctor.
Three people, Al, Bob, and Cary, are standing together at the mall. Al says, “All of us are knaves,” and Bob says, “Exactly one of us is a knight.” What are all three?In a certain kingdom,
There are two people, Ed and Ted. Ed says, “Ted and I are different.” Ted claims, “Only a knave would say that Ed is a knave.” In which category does each belong?In a certain kingdom, there
There are two people, and the first one says, “Either I am a knave or the other person is a knight.” In which category does each belong?In a certain kingdom, there were knights and knaves. The
Given: MIIIUII; Prove: MIIUIIUUse the definitions and postulates given in Example 2 to prove the theorems in Problems 9–14. Give both statements and reasons.
Given: MI; Prove: MIIUIIUUse the definitions and postulates given in Example 2 to prove the theorems in Problems 9–14. Give both statements and reasons.
Given: MI; Prove: MIUIUUse the definitions and postulates given in Example 2 to prove the theorems in Problems 9–14. Give both statements and reasons.
Given: MIII; Prove: MUse the definitions and postulates given in Example 2 to prove the theorems in Problems 9–14. Give both statements and reasons.
The game of soccera. players undefined termb. If a player is fouled, then that player’s team gets a free kick.c. free kick defined termd. goal defined terme. The referee objectively applies the
The game of baseballa. base b. ball c. home run d. If the player accumulates three strikes, then the player is called out. e. If an opposing player catches a foul ball, then the batter is called
The game of Monopoly®a. diceb. luxury taxc. doublesd. If you roll three doubles in a row, then you go directlyto jail.e. jailFor each of the situations in Problems 1–8, classify each item as an
You are given nine steel balls of the same size and color. One of the nine balls is slightly heavier in weight; the others all weigh the same. Using a two-pan balance, what is the minimum number of
You are probably familiar with the game of Monopoly®. Just as in mathematics, a game often has some undefined terms or materials used in the game. In addition, there are definitions used in the game
Every idea of mine that cannot be expressed as a syllogism is really ridiculous. None of my ideas about rock stars is worth writing down. No idea of mine that fails to come true can be expressed as a
When I work a logic problem without grumbling, you may be sure it is one that I can understand. These problems are not arranged in regular order, like the problems I am used to. No easy problem ever
No kitten that loves fish is unteachable. No kitten with a tail will play with a gorilla. Kittens with whiskers always love fish. No teachable kitten has green eyes. Kittens have tails unless they
Nobody who really appreciates Beethoven fails to keep silent while the Moonlight Sonata is being played. Guinea pigs are hopelessly ignorant of music. No one who is hopelessly ignorant of music ever
Dwayne got up at 6:00 a.m. and was watching the sunrise from his bedroom window. After the sun came up, he started working on his toothpick tower in his room. The tower was very fragile. While he was
The detective, Columbo, had just arrived at the scene of the crime and found that the professor had been at the lab working for hours. He seemed to have electrocuted himself and ended up blowing the
If you go to college, then you get a good job.If you get a good job, then you make a lot of money.If you do not obey the law, then you do not make a lot of money.You go to college.
If the government awards the contract to Airfirst Aircraft Company, then Senator Firstair stands to earn a great deal of money.If Airsecond Aircraft Company does not suffer financial setbacks, then
Everyone who is sane can do logic.No lunatics are fit to serve on a jury.None of your sons can do logic.In Problems 49–52, write a valid conclusion.These problems were written by Charles Dodgson,
No ducks waltz.No officers ever decline to waltz.All my poultry are ducks.In Problems 49–52, write a valid conclusion.These problems were written by Charles Dodgson, better known as Lewis Carroll.
All hummingbirds are richly colored.No large birds live on honey.Birds that do not live on honey are dull in color.In Problems 49–52, write a valid conclusion.These problems were written by Charles
Babies are illogical.Nobody is despised who can manage a crocodile.Illogical persons are despised.In Problems 49–52, write a valid conclusion.These problems were written by Charles Dodgson, better
If I am tired, then I cannot finish my homework.If I understand the homework, then I can finishmy homework.In Problems 37–48, form a valid conclusion, using all the premises given for each
If we win first prize, we will go to Europe.If we are ingenious, we will win first prize.We are ingenious.In Problems 37–48, form a valid conclusion, using all the premises given for each argument.
If 2 divides a positive integer N and if N is greater than 2, then N is not a prime number.N is a prime number.In Problems 37–48, form a valid conclusion, using all the premises given for each
If I eat that piece of pie, I will get fat.I will not get fat.In Problems 37–48, form a valid conclusion, using all the premises given for each argument. Give reasons.
If a nail is lost, then a shoe is lost.If a shoe is lost, then a horse is lost.If a horse is lost, then a rider is lost.If a rider is lost, then a battle is lost.If a battle is lost, then a kingdom
If we interfere with the publication of false information, we are guilty of suppressing the freedom of others.We are not guilty of suppressing the freedom of others.In Problems 37–48, form a valid
If a · b = 0, then a = 0 or b = 0.a · b = 0In Problems 37–48, form a valid conclusion, using all the premises given for each argument. Give reasons.
a = 0 or b = 0a ≠ 0In Problems 37–48, form a valid conclusion, using all the premises given for each argument. Give reasons.
If you climb the highest mountain, then you feel great.If you feel great, then you are happy.In Problems 37–48, form a valid conclusion, using all the premises given for each argument. Give reasons.
If we go to the concert, then we are enlightened.We are not enlightened.In Problems 37–48, form a valid conclusion, using all the premises given for each argument. Give reasons.
If I am idle, then I become lazy.I am idle.In Problems 37–48, form a valid conclusion, using all the premises given for each argument. Give reasons.
If you learn mathematics, then you are intelligent.If you are intelligent, then you understand human nature.In Problems 37–48, form a valid conclusion, using all the premises given for each
Prove that is an invalid argument. What is the name of this fallacy? b~ [d~ (bd)]
Show that is an invalid argument by constructing a truth table. What is the name of this fallacy? ( b) [( d) V (bd)]
Prove by constructing a truth table.What is the name we give to this type of reasoning? d~ [b~ (bd)]
If the crime occurred after 4:00 a.m., then Smith could not have done it.If the crime occurred at or before 4:00 a.m., then Jones could not have done it.The crime involved two persons, if Jones did
If Missy uses Smiles toothpaste, then she has fewer cavities. Therefore, if Missy has fewer cavities, then she uses Smiles toothpaste.Determine whether each argument in Problems 9–28 is valid or
If the San Francisco 49ers lose, then the Dallas Cowboys win.If the Dallas Cowboys win, then they will go to the Super Bowl.Therefore, if the San Francisco 49ers lose, then the DallasCowboys will go
If Alice drinks from the bottle marked “poison,” she willbecome sick.Alice does not drink from a bottle that is marked “poison.”Therefore, she does not become sick.Determine whether each
If Congress appropriates the money, the project can be completed.Congress appropriates the money.Therefore, the project can be completed.Determine whether each argument in Problems 9–28 is valid or
If I don’t get a raise in pay, I will quit.I don’t get a raise in pay.Therefore, I quit.Determine whether each argument in Problems 9–28 is valid or invalid. If valid, name the type of
All cats are animals.This is not an animal.Therefore, this is not a cat.Determine whether each argument in Problems 9–28 is valid or invalid. If valid, name the type of reasoning and if invalid,
If a2 is even, then a must be even.a is odd.Therefore, a2 is odd.Determine whether each argument in Problems 9–28 is valid or invalid. If valid, name the type of reasoning and if invalid, determine
a.b.Determine whether each argument in Problems 5–8 is valid or invalid. Give reasons for your answer. b~:: d~ b-d
There are similarities between Pólya’s problem-solving method and the steps a detective will go through in solving a case. Rewrite each of these detective problem-solving steps using the language
Formulate a conclusion for each argument.a. If you attend class, then you will pass the course.If you pass the course, then you will graduate.b. If you graduate, then you will get a good job.If you
Formulate a conclusion for each statement by using indirect reasoning.a. If the cat takes the rat, then the rat will take the cheese. The rat does not take the cheese.b. If x is an even number, then
Use direct reasoning to formulate a conclusion for each of the given arguments.a. If you play chess, then you are intelligent. You play chess.b. If x + 2 = 3, then x = 1.x + 2 = 3.c. If you are a
Translate into symbolic form: Either Alfie is not afraid to go, or Bogie and Clyde will have lied.
The contract states, “The tenant shall not let or sublet the whole or any portion of the premises to anyone for any purpose whatsoever, unless written permission from the landlord is obtained.”
The contract states, “No alterations, redecorating, tacks, or nails may be made in the building, unless written permission is obtained.” Write this statement symbolically.
An airline advertisement states, “OBTAIN 40% OFF REGULAR FARE.”Read the fine print: “You must purchase your tickets between January 5 and February 15 and fly round trip between February 20 and
To qualify for a loan of $200,000, an applicant must have a gross income of $72,000 if single, or $100,000 combined income if married. It is also necessary to have assets of at least $50,000. Write
The following article is by N. Gregory Mankiw, “First Principles,” Fortune, January 24, 2000, p. 36. Discuss the article’s assertion. Why Getting Married Still Costs You If a man and woman have
2x + 3y = 8 if x = 1 and y = 2.Write the negation of each compound statement in Problems 41–54.
If x = –5, then x2 = 25.Write the negation of each compound statement in Problems 41–54.
If x – 5 = 4, then x = 1.Write the negation of each compound statement in Problems 41–54.
If x + 2 = 5, then x = 3.Write the negation of each compound statement in Problems 41–54.
If you’re out of Schlitz, you’re out of beer.Write the negation of each compound statement in Problems 41–54.
If I can’t go with you, then I’ll go with Bill.Write the negation of each compound statement in Problems 41–54.
Missy is not on time and she missed the boat.Write the negation of each compound statement in Problems 41–54.
Theron is not here and he is not at home.Write the negation of each compound statement in Problems 41–54.
Stephanie went to the basketball game or to the soccer game.Write the negation of each compound statement in Problems 41–54.
Cole went to Macy’s or Sears.Write the negation of each compound statement in Problems 41–54.
∼p → ∼qWrite the negation of each compound statement in Problems 41–54.
∼p → qWrite the negation of each compound statement in Problems 41–54.
p → ∼qWrite the negation of each compound statement in Problems 41–54.
p → qWrite the negation of each compound statement in Problems 41–54.
Prove (p → q) ⇔ (~p ∨ q) .
In the text we used laws of logic to prove that (b~ d) (bd)~ Use a truth table to prove this result.
Prove De Morgan’s law: b~ d~ (b / d)~ v >
Prove the law of contraposition by using a truth table. Table 3.9 Additional Operators Either p or q p q (pVg)^~(p^q) TT F TF T FT T F F F Neither p nor q p unless q ~(pv q) ~qp F T F T F T T F p
Prove the law of double negation by using a truth table. Table 3.9 Additional Operators Either p or q p q (pVg)^~(p^q) TT F TF T FT T F F F Neither p nor q p unless q ~(pv q) ~qp F T F T F T T F p
Show that the definition for neither p nor q could also be ~p ~q.

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