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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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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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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
(
)
1−x - - + +
(
)
4−x - - - +
(
)
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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