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Translate each of the sentences in Exercises 1-26 into a statement in the predicate calculus. (Red letters are to be used for the relevant predicates or terms where appropriate.)
1. | Every good girl deserves fruit. |
2. | Good boys deserve fruit always. |
3. | All cows eat grass. |
4. | No cows eat grass. |
5. | Some cows eat grass. |
6. | Some birds are fishes. |
7. | Some cows are not birds and some are. |
8. | Some cows are birds but no cows are fishes. |
9. | Although some city drivers are insane, Dorothy is a very sane city driver. |
10. | Even though all mathematicians are nerds, Waner and Costenoble are not nerds. |
11. | If one or more lives are lost, then all lives are lost. |
12. | If every creature evolved from lower forms, then you and I did as well. |
13. | Some numbers are larger than two; others are not.. |
14. | Every number smaller than 6 is also smaller than 600 |
In Exercises 15-26, you can use the convention that the letters i through n represent positive integers.
15. | 12 is divisible by 6. |
16. | 13 is not divisible by 6. |
17. | For any positive integer m, if 12 is divisible by m, then so is 24. |
18. | If 13 is not divisible by m, then neither is 17. |
19. | 15 is divisible by some positive integer. |
20. | 15 is divisible by a positive integer other than 15 or 1. |
21. | 17 is prime (that is, not divisible by any positive integer except itself and 1). |
22. | 15 is not prime. (See (21).) |
23. | There is no smallest positive real number. (Use the convention that the letters x through z represent real numbers.) |
24. | There is no largest positive integer. |
25. | If 1 has property P, and if (n+1) has property P whenever n does, then every positive integer has property P. (This statement is called the Principle of Mathematical Induction.) |
26. | If 2 has property P, and if (n+2) has property P whenever n does, then every even positive integer has property P. |
Translate the statements in Exercises 27-34 into words.
27. | x[RxSx]; R = "is a raindrop," S = "makes a splash." |
28. | y[CyMy]; C = "is a cowboy," M = "is macho." |
29. | z[DzWz]; D = "is a dog," W = "whimpers." |
30. | z[Dz~Wz]; D = "is a dog," W = "whimpers." |
31. | x[Dx~Wx]; D = "is a dog," W = "whimpers." |
32. | ~x[DxWx]; D = "is a dog," W = "whimpers." |
33. | z,y[CzCyWz~Wy]; C = "is a cat," W = "whimpers" |
34. | x[Px y[PyL(x,y)]], P = "is a person," L(x,y) = "y is older than x." |
Communication and Reasoning Exercises
35. | The claim that every athlete drinks ThirstPro is false. In other words, no athletes drink ThirstPro, right? |
36. | Give one advantage that predicate calculus has over propositional calculus. |
37. | Your friend claims that the quantifiers and are insufficient for her purposes; she requires new quantifiers to express the phrases "for some" and "there does not exist". How would you respond? |
38. | Consider a new quantifier, "" meaning "for no" (as in "for no x can x be larger than itself") Express in terms of the quantifiers you already have?. |