Unformatted text preview: eeking is the fact that if t and x are any numbers in the interval p, 1
then
1 x t 
1 x t 
1
x
tx
t
p2
The latter expression will be less than 1 when x t  p 2 . With this observation we define p 2
and we have found a number with the required properties.
10. a. If either 0 or 2 then
arctan tan /2 arctan tan /2 2 . We have two cases to consider:
Case 1: Suppose that 0 . In this case we have 0 and also
2
0 .
2
2
Therefore
arctan tan /2 arctan tan /2 .
2
2
2
Case 2: Suppose that 2. Since
2
2
we see that
arctan tan
2
2
and since
3
2
2
2
and so
arctan tan /2
2
Therefore
arctan tan /2 arctan tan /2 .
2
2
2 27 b. Ask Scientific Notebook to solve the equation
arctan tan /2 arctan tan /2 2 . Are you satisfied with the answer that it gives?
11. If x is any rational number then
lim lim cos n!x m nÝmÝ 1.
p We suppose that x is rational and we express x in the form q where p and q are integers and q 0.
Whenever an integer n is greater than q, the number n! x is an even integer and for such integers
n we have
lim cos n!x m m Ý 1 1
lim
mÝ
Since lim m Ý cos n!x m 1 whenever n is sufficiently large we have
lim lim cos n!x m 1. lim lim cos n!x m 0. nÝmÝ 12. If x is any irrational number then
nÝmÝ We suppose that x is irrational. Given any integer n we know from the fact that n! x is not an
integer that
 cos n!x  1
and therefore that
lim cos n!x m 0.
mÝ
Therefore
lim lim cos n!x nÝmÝ m 0. 4 Elements of Set Theory
Exercises on Set Notation
1. Given objects a, b and y and given that a, b a, y , prove that b y. Solution: Since y a, y and a, b a, y , we know that y
a, b . We know, therefore, that
either y a or y b. However, if y a then the equation a, b a, y becomes a, b a and we
deduce from the fact that b
a that b a. So in this case too we have y b. 2. Prove that if a, b, x and y are any given objects and if a, b x, y then either a x and b y, or a y
and b x. Solution: Since a a, b and a, b x, y we know that a
x, y . Therefore either a x or
a y. In the event that a x we have a, b a, y and it follows from Exercise 1 that b y. Similarly,
if a y then b x. 3. Prove that if a, b, x and y are any given objects and if
a , a, b
then a x and b y. 28 x , x, y , The preceding exercises guarantee that if
a , a, b x , x, y ,
then either a x or a, b x, y from which we can see at once that a x and b y.
4. Describe the set p . Solution: Since the only subset of the set
5. Describe the set p p is itself we have p . . Hint: You should be able to prove that p p , . 6. Given that A a, b, c, d , list all of the members of the set p A . Solution: You should be able to show that p a, b, c, d is , a , b , c , d , a, b , a, c , a, d , b, c , b, d , c, d , a, b, c , a, b, d , a, c, d , b, c, d , a, b, c, d
7. Given tha...
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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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