Study Guides (256,509)
CA (124,674)
UTSG (8,518)
MAT (561)
MAT137Y1 (82)
None (3)

Summary notes 1

33 Pages
525 Views

Department
Mathematics
Course Code
MAT137Y1
Professor
None

This preview shows pages 1-3. Sign up to view the full 33 pages of the document.
MAT137Y1a.doc
Page 1 of 33
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
www.notesolution.com
MAT137Y1a.doc
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
www.notesolution.com
MAT137Y1a.doc
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
www.notesolution.com

Loved by over 2.2 million students

Over 90% improved by at least one letter grade.

Leah — University of Toronto

OneClass has been such a huge help in my studies at UofT especially since I am a transfer student. OneClass is the study buddy I never had before and definitely gives me the extra push to get from a B to an A!

Leah — University of Toronto
Saarim — University of Michigan

Balancing social life With academics can be difficult, that is why I'm so glad that OneClass is out there where I can find the top notes for all of my classes. Now I can be the all-star student I want to be.

Saarim — University of Michigan
Jenna — University of Wisconsin

As a college student living on a college budget, I love how easy it is to earn gift cards just by submitting my notes.

Jenna — University of Wisconsin
Anne — University of California

OneClass has allowed me to catch up with my most difficult course! #lifesaver

Anne — University of California
Description
MAT137Y1a.doc Lecture #1 Tuesday, September 9, 2003 S ETS non-example: A ={paintings that are be}utiful example: A ={natural numbers divisibl} by 4 Notation 4 A 4 is in A 3 A 4 is not in A B ={naturalnumbersdivisible}y7 AB = {aturalnumbersdivisibleby4or}7 union of A and B = 4,7,8,12,14,...} AB = naturalnumbersdivisibleby4and}7 intersection of A and B FACTS A BOUT R EAL N UMBERS N = natural numbers 1,2,3,4}... Z = integers {...,3,2,1,0,1}2,3,... Q = rational numbers{1,2,3,4}... R = real numbers = all this and more (+ irrational numbers) Geometrically -5 -4 -3 -2 -1 0 1 2 3 4 5 The Real Line any point on the line represents a real number Intervals 1,2]={x R :1 x } 1,2)={x R :1< x < } 0,) Ordering (Inequalities) a < b If a, b are real numbers, then exactly one of the folla = bis true: a > b Important Properties If a < b and c > , then ac < bc If a < b and c < , then ac > bc Page 1 of 33 www.notesolution.com MAT137Y1a.doc DistanceAbsolute Values a =distance fromato 0= a,ifa 0 a,ifa< 0 ab = distancebetween a andb Triangle Inequality a+b a + b analogous to triangle theorem from geometry a b c c a +b c Lecture #2 Thursday, September 11, 2003 T RIANGLE NEQUALITY a+b a + b,a,bR Proof (A Proof By Cases) 1) If a,b both 0 Then a +b 0 a+b = a+b = a + b 2) If a > 0,b < 0, a b Then a +b 0 a+b = a+b a + b = ab Because b < 0 , so a +b < a b 3) If Then a +b 0 a+b = a+b = a + b 4) a > 0,b < 0, a < b 5) 6) Page 2 of 33 www.notesolution.com MAT137Y1a.doc R EVIEW OF INEQUALITIES Solve 2 x+ 6 2x + 3) x1 )x4 )x+2 )> 0 2 x 3 2) 2x 2 Where is it = 0? x =1,4,2 3 x 1 1) x 2 x [1,) -2 1 4 x , 3 - - + + 2 (x1 ) (x4 ) - - - + (x+2 ) - + + + Product - + - + x (2,) (4,) Inequalities With Absolute Values x - 0 < 0 < x < it means thax (, ) x xc < c - c c + it means that < x < c+ x c,c+ ) x >5 x > 5 -2 3 8 x < 5 ( 2 )8,( ) C OORDINATE G EOMETRY Rectangular Coordinates y-axis 3 2 1 (3,1) 1 2 3 x-axis Page 3 of 33 www.notesolution.com
More Less
Unlock Document


Only pages 1-3 are available for preview. Some parts have been intentionally blurred.

Unlock Document
You're Reading a Preview

Unlock to view full version

Unlock Document

Log In


OR

Don't have an account?

Join OneClass

Access over 10 million pages of study
documents for 1.3 million courses.

Sign up

Join to view


OR

By registering, I agree to the Terms and Privacy Policies
Already have an account?
Just a few more details

So we can recommend you notes for your school.

Reset Password

Please enter below the email address you registered with and we will send you a link to reset your password.

Add your courses

Get notes from the top students in your class.


Submit