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MAT137Y1a.doc
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Lecture #1 Tuesday, September 9, 2003
SETS
non-example:
{
}
beautiful are that paintings=A
example:
{
}
4by divisible numbers natural=A
Notation
A
4
4 is in A
A
3 4 is not in A
{
}
7by divisible numbers natural=B
{
}
{ }
,...14,12,8,7,4
7or 4by divisible numbers natural
=
=
BA union of A and B
{
}
7 and 4by divisible numbers natural=BA intersection of A and B
FACTS ABOUT REAL NUMBERS
N = natural numbers =
{
}
,...4,3,2,1
Z = integers =
{
}
,...3,2,1,0,1,2,3...,
Q = rational numbers =
{
}
,...4,3,2,1
R = real numbers = all this and more (+ irrational numbers)
Geometrically
any point on the line represents a real number
Intervals
[
]
{
}
21:2,1=xx R
(
)
{
}
21:2,1<<=xx R
[
)
,0
Ordering (Inequalities)
If a, b are real numbers, then exactly one of the following is true:
>
=
<
ba
ba
ba
Important Properties
If ba < and 0
>
c, then bcac<
If ba < and 0
<
c, then bcac>
0
1
2
3
4
5
-
1
-
2
-
3
-
4
-
5
The Real Line
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Page 2 of 33
Distance/Absolute Values
<
== 0 if ,
0 if ,
0 to from distance aa
aa
aa
baba and between distance=
Triangle Inequality
baba ++
analogous to triangle theorem from geometry
bac+c
Lecture #2 Thursday, September 11, 2003
TRIANGLE INEQUALITY
R++ bababa ,,
Proof (A Proof By Cases)
1) If 0both , ba
Then 0+ ba
bababa +=+=+
2) If baba <> ,0,0
Then 0+ ba
baba
baba
=+
+=+
Because 0<b, so baba <+
3) If
Then 0+ ba
bababa +=+=+
4) baba <<> ,0,0
5)
6)
a b
c
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Page 3 of 33
REVIEW OF INEQUALITIES
Solve
1)
+
2
3
,
2
3
32
632
x
x
x
x
2)
[
)
+
,1
1
22
532
x
x
x
x
3)
(
)
(
)
(
)
0241 >+xxx
Where is it = 0? 2,4,1
=
x
(
)
1x - - + +
(
)
4x - - - +
(
)
2+x - + + +
Product - + - +
(
)
(
)
,41,2x
Inequalities With Absolute Values
δ
<x it means that
( )
δδ
δ
δ
,
0
<
<
x
δ
<cx it means that
( )
δδ
δ
δ
+
+
<
<
ccx
cxc
,
( ) ( )
<
>
>
,82,
53
53
53
x
x
x
COORDINATE GEOMETRY
Rectangular Coordinates
8
-
2
3
c
+
c
x
c
-
4
-
0
x
x-axis
y-axis
1
2
3
1
2
3
(3,1)
-
2
1
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