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Unformatted text preview: told me that it has never rained on a day on which you have remembered to take your umbrella. Solution: You have not told me that it has never rained on a day on which you have remembered to
take your umbrella.
12. You lied when you told me that it has never rained on a day on which you have remembered to take your
umbrella. Solution: You did not lie when you told me that it has never rained on a day on which you have
remembered to take your umbrella.
13. I was joking when I said that you lied when you told me that it has never rained on a day on which you have
remembered to take your umbrella. Solution: I was not joking when I said that you lied when you told me that it has never rained on a
day on which you have remembered to take your umbrella.
14. This, Watson, if I mistake not, is our client now. Solution: This statement is meaningless because it refers to itself. We should have been able to
expect better logical precision from Sherlock Holmes.
15. a. For every real number x there exists a real number y such that
5y 4x
2x 2 xy y 2
2
.
3x y
3 x 2 xy y 2
x3 y3
Is this statement is true? Solution: Yes this statement is true.
Negation: There exists a real number x such that for every real number y, the equation
5y 4x
2x 2 xy y 2
2
3x y
3 x 2 xy y 2
x3 y3
is either false or meaningless.
b. There exists a real number x such that for every real number y we have
5y 4x
2x 2 xy y 2
2
.
3
3
2 xy y 2
3x y
3x
x
y
Is this statement is true? Solution: This statement is false because the equation shown her can’t be true when y x.
Negation: For every real number x there exists a real number y for which the equation
5y 4x
2x 2 xy y 2
2
3x y
3 x 2 xy y 2
x3 y3
is either false or meaningless.
c. For every real number x and every real number y x we have
5y 4x
2x 2 xy y 2
2
.
3
3
2 xy y 2
3x y
3x
x
y
Is this statement is true? 5 Solution: This statement is true.
Negation: There exists a real number x and there exists a real number y
5y 4x
2x 2 xy y 2
2
3x y
3 x 2 xy y 2
x3 y3 x such that the equation is either false or meaningless. Some Exercises on The Use of Conditionals
In the exercises that follow you should assume that P, Q, R and S are given statements that may be either true
or false.
1. Write down the denial, the converse and the contrapositive form of each of the following statements:
a. All cats scratch.
The converse: All things that scratch are cats.
The contrapositive: All things that do not scratch are not cats.
The denial: There is at least one cat that does not scratch.
b. If what you said yesterday is correct, then Jim has red hair.
The converse: If Jim has red hair then what you said yesterday is correct.
The contrapositive: If Jim does not have red hair then what you said yesterday is false.
The denial: What you said yesterday is correct and Jim does not have red hair.
c. If a triangle ABC has a right angle at C then
AB 2 AC 2 BC 2 . The converse: Every triangle A...
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This note was uploaded on 11/26/2012 for the course MATH 2313 taught by Professor Staff during the Fall '08 term at Texas El Paso.
 Fall '08
 STAFF
 Math, Calculus

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