1873_solutions

M it is possible to find an integer n such that mn is

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Unformatted text preview: le to find an integer n such that for every odd integer m the number mn is even. Solution: These two statements do not say the same thing but they are both true. 9. a. For every number x it is possible to find a number y such that xy  0. b. It is possible to find a number y such that for every number x we have xy  0. Solution: These two statements do not say the same thing but they are both true. 3 10. a. For every number x it is possible to find a number y such that xy 0. b. It is possible to find a number y such that for every number x we have xy 0. Solution: These two statements do not say the same thing but they are both false. 11. a. For every number a and every number b there exists a number c such that ab  c. Solution: True b. For every number a there exists a number c such that for every number b we have ab  c. Solution: False 12. a. For every number a and every number c there exists a number b such that ab  c. b. For every number a there exists a number b such that for every number c we have ab  c. Solution: These two statements do not say the same thing but they are both false. 13. a. For every nonzero number a and every number c there exists a number b such that ab  c. Solution: True b. For every nonzero number a there exists a number b such that for every number c we have ab  c. Solution: False Some Exercises on Negations and the Quantifiers Write a negation for each of the following statements: 1. All roses are red. Solution: The negation could say: Not all roses are red. Alternatively it could say: There is at least one rose that isn’t red. However it would be wrong to write the negation as: All roses are not red. 2. In Sam’s flower shop there is at least one rose that isn’t red. Solution: All the roses in Sam’s flower shop are red. 3. In every flower shop there is at least one rose that isn’t red. Solution: There is at least one flower shop in which all the roses are red. 4. I believe that all roses are red. Solution: I do not believe that all roses are red. 5. There is at least one person in this room who thinks that all roses are red. Solution: No one in this room thinks that all roses are red. 6. Every person in this room believes that all roses are red. There is at least one person in this room who does not believe that all roses are red. 7. At least half of the people in this room believe that all roses are red. Solution: More than half of the people in this room do not believe that all roses are red. 8. Every man believes that all women believe that all roses are red. Solution: There are men who do not believe that all women believe that all roses are red. 4 9. You were at least an hour late for work every day last week. Solution: There was at least one day last week on which you began work less than an hour late. 10. It has never rained on a day on which you have remembered to take your umbrella. Solution: There have been days on which it has rained and on which you have remembered to take your umbrella. 11. You...
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