2. In this question, you are asked to model connections between actors using predicate logic. Let be the domain of every actor, alive or dead, who has ever been in any movie. Let C be the co-star predicate on pairs of actors from D, defined by C(x, y) = x and y have been in a movie together. The 'Bacon game' is a game where players try to connect individuals to the actor Kevin Bacon by using the co-star predicate C. The Bacon number of an actor x (that is, where x is an element of D) is the smallest number of links needed, using the co-star predicate C, to connect x to Kevin Bacon. For example, Daniel Kaluuya was in the movie Sicario with Jeffrey Donovan, who in turn was in Sleepers with Kevin Bacon. So, Daniel Kaluuya has a Bacon number of 2. (This example was generated by oracleofbacon.org, where you can determine the Bacon number for many other actors.) Let b₁ = D be Kevin Bacon, i.e., let b, be the special name that we give to Kevin Bacon, given the unique nature of his role in the game. a. Write a predicate B₂(x) in one variable that reflects that the Bacon number of an actor x € D is at most two. By the definition of the Bacon number, this means that there is some actor y such that x and y were in a movie together and that y and bo were in a movie together. The expression for the predicate must involve a quantifier. b. Similarly, write a predicate B₂(x) in one variable that reflects that the Bacon number of an actor x is at most three. This will likely involve multiple quantifiers. c. Notice for B₂ (x), the predicate reflects that the Bacon number of x is at most 3. The predicate does not enforce that the Bacon number is exactly 3, since there is a possibility that the actor x is, e.g., in a movie with b₁, or has a Bacon number of 2 through some other links. Write a predicate for an actor x E D to have Bacon number exactly three, called B₂ (x), by excluding these other possibilities. This predicate should be defined by using C, B and B₂. Be sure to exclude the possibility that x is Kevin Bacon. d. Write a full expression for B₂ (x) that does not use B₂ or B₂, but still uses C. Make sure to parenthesize appropriately. (Hint: use your solution for part c. and negate as necessary.) Write a short (1-2 sentence) explanation of the entire predicate definition in natural language, verifying that it makes sense.

2. In this question, you are asked to model connections between actors using predicate logic. Let be the domain of every actor, alive or dead, who has ever been in any movie. Let C be the co-star predicate on pairs of actors from D, defined by C(x, y) = x and y have been in a movie together. The 'Bacon game' is a game where players try to connect individuals to the actor Kevin Bacon by using the co-star predicate C. The Bacon number of an actor x (that is, where x is an element of D) is the smallest number of links needed, using the co-star predicate C, to connect x to Kevin Bacon. For example, Daniel Kaluuya was in the movie Sicario with Jeffrey Donovan, who in turn was in Sleepers with Kevin Bacon. So, Daniel Kaluuya has a Bacon number of 2. (This example was generated by oracleofbacon.org, where you can determine the Bacon number for many other actors.) Let b₁ = D be Kevin Bacon, i.e., let b, be the special name that we give to Kevin Bacon, given the unique nature of his role in the game. a. Write a predicate B₂(x) in one variable that reflects that the Bacon number of an actor x € D is at most two. By the definition of the Bacon number, this means that there is some actor y such that x and y were in a movie together and that y and bo were in a movie together. The expression for the predicate must involve a quantifier. b. Similarly, write a predicate B₂(x) in one variable that reflects that the Bacon number of an actor x is at most three. This will likely involve multiple quantifiers. c. Notice for B₂ (x), the predicate reflects that the Bacon number of x is at most 3. The predicate does not enforce that the Bacon number is exactly 3, since there is a possibility that the actor x is, e.g., in a movie with b₁, or has a Bacon number of 2 through some other links. Write a predicate for an actor x E D to have Bacon number exactly three, called B₂ (x), by excluding these other possibilities. This predicate should be defined by using C, B and B₂. Be sure to exclude the possibility that x is Kevin Bacon. d. Write a full expression for B₂ (x) that does not use B₂ or B₂, but still uses C. Make sure to parenthesize appropriately. (Hint: use your solution for part c. and negate as necessary.) Write a short (1-2 sentence) explanation of the entire predicate definition in natural language, verifying that it makes sense.

Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018

18th Edition

ISBN:9780079039897

Author:Carter

Publisher:Carter

Chapter10: Statistics

Section10.6: Summarizing Categorical Data

Problem 13CYU

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