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- 38. Prove or disprove that .11. (See Exercise 10.) According to Definition 5.29, is defined in by if and only if . Show that if and only if . 10. An ordered field is an ordered integral domain that is also a field. In the quotient field of an ordered integral domain define by . Prove that is a set of positive elements for and hence, that is an ordered field. Definition 5.29 Greater than Let be an ordered integral domain with as the set of positive elements. The relation greater than, denoted by is defined on elements and of by if and only if . The symbol is read “greater than.” Similarly, is read “less than.” We define if and only if. As direct consequences of the definition, we have if and only if and if and only if . The three properties of in definition 5.28 translate at once into the following properties of in . If and then . If and then . For each one and only one of the following statements is true: . The other basic properties of are stated in the next theorem. We prove the first two and leave the proofs of the others as exercises.Suppose in a small town there are three places to eat, a Chineserestaurant, a Mexican restaurant, and a pizza place. Everyone in town eats dinnerin one of these places or has dinner at home. Assume that 20% of those who eat inthe Chinese restaurant go to Mexican next time, 20% eat home and 30% go to thepizza place. From those who eat in the Mexican restaurant, 10% go to the pizzaplace, 25% go to the Chinese restaurant, and 25% eats at home next time. Fromthose who eat at the pizza place, 30% eat at home, 30% eat at the Chineserestaurant, and 10% eat at the Mexican restaurant next time. Those who eat athome, 20% go to the Chinese restaurant, 25% go to Mexican restaurant, and 30% tothe pizza place. a. Set us the matrix of transition probabilities and show the transition b. Find the steady state probabilities c. In the long run, which restaurant will have the most customer
- Suppose in a small town there are three places to eat, a Chineserestaurant, a Mexican restaurant, and a pizza place. Everyone in town eats dinnerin one of these places or has dinner at home. Assume that 20% of those who eat inthe Chinese restaurant go to Mexican next time, 20% eat home and 30% go to thepizza place. From those who eat in the Mexican restaurant, 10% go to the pizzaplace, 25% go to the Chinese restaurant, and 25% eats at home next time. Fromthose who eat at the pizza place, 30% eat at home, 30% eat at the Chineserestaurant, and 10% eat at the Mexican restaurant next time. Those who eat athome, 20% go to the Chinese restaurant, 25% go to Mexican restaurant, and 30% tothe pizza place. a. Set us the matrix of transition probabilities and show the transition diagram b. Find the steady state probabilities c. In the long run, which restaurant will have the most customerIf all fruit bearing plants are flowering plants, and all flowering plants attracti nsects. Which of the following is a valid conclusion? a. Some fruit bearing plants do not attract insects. b. If a plant attracts insects, then it is fruit bearing. c. All fruit bearing plants attract insects. d. All flowering plants are fruit bearing. e. Some flowering plants are fruit bearing.It was found that 33% of all diners order vegan meals and that 90% of all diners are undergraduate students. Furthermore, 30% of all undergraduate students order vegan meals. Label the event “diner orders a vegetarian meal” as V and call the event “diner is an undergraduate student” as U. 1. Are V and U independent? 2. Are V and U mutually exclusive? 3. Are V and U collectively exhaustive?
- How would you write a formal proof for if AnB = B then B is a subset of A?Somie, a leader of the underworld, was killed by one of his own band of four henchmen. Detective Sharpinterviewed the men and determined that all were lying except for one. He deduced who killed Somieon the basis of the following statements:a. Socko: Lefty killed Somie.b. Fats: Muscles didn’t kill Somie.c. Lefty: Muscles was shooting craps with Socko when Somie was knocked off.d. Muscles: Lefty didn’t kill Somie.Construct a Venn diagram to answer the question. Doctor X and Doctor Y have 24 patients. 12 patients have fever, 6 patients have colds, 15 patients have headache. 1 patient has all three sicknesses. 2 patients have fever and colds, but no headache. 2 patients have colds and headache, but no fever. If all the patients of Doctor X and Doctor Y have at least one of the mentioned sickness, how many patients have fever and headache, but no colds?
- According to a recent survey of first-year high school students, 28% chew gum daily. The students were also asked if they had recently gotten a cavity filled at the dentist. Of the 47% of first-year students who responded that they had a cavity recently filled, only 39% chewed gum daily. Is chewing gum independent of having a cavity filled recently? Yes, P(Gum) = P(Gum|Cavity). Yes, P(Gum) = P(Cavity|Gum). No, P(Gum) ≠ P(Gum|Cavity). No, P(Gum) ≠ P(Cavity|Gum).Determine whether the given set is basis for M2,2 * same problem split in 2Can you prove that A ∩ B = B ∩ A for every two sets A and B without using the commutative law of conjunction of two statements? Just simply?