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Unformatted text preview: rtion that 57 is prime for every such integer is true. The fact that the number 57 happens not to be prime has no bearing
on this exercise. Exercises on Order of Appearance of Unknowns
For each of the following pairs of statements, decide whether or not the statements are saying the same thing.
Except in the first two exercises, say whether or not the given statements are true.
1. a. Every person in this room has seen a good movie that has started playing this week.
b. A good movie that has started playing this week has been seen by every person in this room. Solution: The second statement asserts that there is one particular good movie that everyone in
the room has seen. The first statement says less. It asserts that everyone has seen a good movie but
leaves open the possibility that different people may have seen different movies.
2. a. Only men wearing top hats may enter this hall. 2 b. Only men may enter this hall wearing top hats.
c. Men wearing top hats only may enter this hall.
d. Men wearing only top hats may enter this hall.
e. Men wearing top hats may enter this hall only. Hint: These five statements are all different from one another. The first statement is ambiguous. It
could mean that the only people who may enter this hall are men who are wearing top hats.
However, with a different voice inflection it could mean that top hats are permitted only to men and
that women will not be permitted entry if they are wearing top hats, leaving open the question of
whether a woman who is not wearing a top hat may enter the hall. In other words, a change in
voice inflection could make statements a. and b. say the same thing.
3. a. For every nonzero number x there is a number y such that xy 1. Solution: This statement is true because of x is any nonzero number then we have x 1/x 1. b. There is a number y for which the equation xy 1 is true for every nonzero number x. Solution: This statement is false.
4. a. For every number x 0, 1 there exists a number y 0, 1 such that x y. Solution: True
b. There is a number y 0, 1 satisfying x y for every number x 0, 1 . Solution: False
5. 0, 1 such that x y. a. For every number x 0, 1 there exists a number y b. There is a number y 0, 1 satisfying x y for every number x 0, 1 . Solution: These two statements are not saying the same thing but who cares! Both statements are
false.
6. a. For every number x 0, 1 there exists a number y This statement is true. Given any x
y
0, 1 such that x y.
b. There is a number y 0, 1 satisfying x 0, 1 such that x y. 0, 1 , if we define y x then we have found a number
y for every number x 0, 1 . This statement is false because it asserts that the interval 0, 1 has a largest member.
7. a. For every number x 0, 1 there exists a number y b. There is a number y 0, 1 satisfying x 0, 1 such that x y for every number x y. 0, 1 . This time, both of the statements are true.
8. a. For every odd integer m it is possible to find an integer n such that mn is even.
b. It is possib...
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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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