1.2. SETS AND EQUIVALENCE RELATIONS 9
for (:13, y) in R2. This is actuallr matrix multiplication; that is,
a o :1: _ as: + by
c d y _ es: + dy '
Maps from R to Rm given by matrices are called linear m
10 CHAPTER 1. PRELIMINARIES
Example 1.16. The function f (:12) = 3:3 has inverse f"1(:13) = 3/5 by
Example 1.12.
Example 1.17. The natural logarithm and the exponential functions,
f($) = 111:1: and f_
2.1. MATHEMATICAL INDUCTION 19
result is true for n greater than or equal to 1. Then
(a + Inn = (a + b)(a + b)
= (a + b) (g: (E) akink)
_ n n n+1 1114: H n sn+1s
(19:1 5 +Z(k)ab

S:
E3
+
H
+
M:
PS"
1 .3. EXERCISES 15
(a) Dene a function f : N :> N that is onetoone but not onto.
(b) Dene a function f : N > N that is onto but not onetoone.
21. Prove the relation dened on R2 by (:31, m) N (:32, pg
18 CHAPTER 2. THE INTEGERS
Example 2.2. For all integers n. 33 3, 2 3:2 n.  4. Since
8 = 23 :> 3 + 4 = 7.
the statement is true for no = 3. Assume that 2" .2: k+4 for k 2 3. Then
2H1 : 2  2 :> 20.
12 CHAPTER 1. PRELIMINARIES
then A m B since PAP"1 = B for
he 2)
Let I be the 2 X 2 identity matrix; that is,
1 0
1_(0 I).
Then IAI"1 = IAI = A; therefore, the relation is reexive. To show
symmetry,
14 CHAPTER 1. PRELIMINARIES
(a) AHB (c) AUB
(b)BnO (d)A(BUC)
2. If A = cfw_(1, b, c, B = cfw_1,2,3, 0 = cfw_1:, and D = [3, list all of the
elements in each of the following sets.
(e)A><B (o)A><B><C
(
1 .3. EXERCISES 13
Example 1.27. In the equivalence relation in Example 1.21, two pairs
of integers, (p, q) and (r, s), are in the same equivalence class when they
reduce to the same fraction in its l
1.2. SETS AND EQUIVALENCE RELATIONS 11
Theorem 1.20. A mapping is invertible if and only if it is both oneto
one and onto.
PROOF. Suppose rst that f : A > B is invertible with inverse g : B :>
A. Th
16
CHAPTER 1. PRELIMINARIES
29. (Projective Real Line) Dene a relation on R2 \ cfw_(0, 0) by letting
(331,391) m (332,3;2) if there exists a nonzero real number A such that
(ahgl) : (Jag, Agg). Prove
The Integers
The integers are the building blocks of mathematics. In this chapter
we will investigate the fundamental properties of the integers, including
mathematical induction, the division algor
pie
5
3.14
6
3
31.4 m
78.5 m2
18.84 cm
28.26 cm2
C
A
3 ft/2 radius
1.5
7.065 ft2
9.42 ft
5 yd
5
31.4 yd
78.5 yd2
A
C
C
A
2 diameter twice radius
6.28 ft
3.14 ft2
3.14
11
5
Probility of dart
78.5
379.9
The circle graph shows how a family spends its annual income. If
$23,800
is used for Entertainment and Clothing combined, what is the total annual income?
Solve X
60,000
28
85,000
Solve for
y
.
Err:50
uiz 1r Date Submitted: llefZl?
4. Interpreting a Venn diagram of 2 sets
The Venn diagram shows the memberships for the Soccer Ciub and the Basketbaii CEub.
Use the diagram to answer the questions be