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5
Chapter
1
Speaking Mathematically
3. Given any two real numbers, there is
a real number in
d. If
between.
a. Given any two real numbers a
ber c such that c is
b. For any two _,
_.
_
and
b, there is a real num_
such that c is between
a
any
_,
_
such that s
and b.
c. If an equation is quadraticlthen it
o. It t _, then E
e. For all quadratic equations E,
c.
I
1. Every positive number has
using variables. Determine, as best as you can,
whether the
statements are true or false.
.\
a. There are real numbers
u+u <u-u.
such that
For all real numbers
a
b,
la + bl
<
lal
+
8.
For all objects
"/,
if
a. All squares _.
b. Every square _.
c. If
real number s.
_.
13. There is a real number whose product
with every real number
"/ is a square then ./ has four sides.
equals zero.
.u.
!:*. .
has the property rhat its
real number b,
it
_.
b. There is a real number a such that the pioduct
ofa
c. There is a real number a with
an object is a square, then
number
_.
the
proj..ty that for wery
Answers for Test yourself
1' true; all elements
of a set 2. is true; also has to be true 3. there is
at least one thing
1.2
f,
I
t
The Language of Sefs
' ' ' when we attempt to express in mathematicar symbors a corulition propr.tsed
in words.
First, we mu.st understand thoroughly the condition.
second, we must be familinr with
the forms of mathematical e.rpression. _George polyri
(18g7_19g5)
Use of the word set as a formal mathematical term was
introduced
( I 845- I 9 I 8). For most mathematical
purposes we can think of
cantor
r
a. Some _ has the property that its
b. There is a real number r such that the product
ofr
c. There is a real number r with the p.op..ty
that for eFry
lbl.
In each of 8- I 3, fill in the blanks to rewrite the
given statement.
_.
leaves the number unchanged.
u and u with the property that
and,
positive square root.
12. There is a real number whose
product with every number
b. There is a real number .r such that x2 < x.
c. For all positive integers n, n2 > n.
d.
a
a. AII positive numbers
b. Foranypositive numbe.i there is
for e.
c. For all positive numbers e. there is--positive
7. Rewrite the following statements less formally,
without
s
_.
such that
_,
_,
_.
10. Every nonzero real number
has a reciprocal.
a. All nonzero real numbers
b. For all nonzero real numbe:s r, there is
for r.
c. For all nonzero real numbers r, there is alal
number
The_cube root
ofany negative real number is negative.
negative real number s, the cube rlot of
For any real number s, if s is
then
If a real number s
then
has at most two
a. All quadratic equations
b. Every quadratic equation-
> r.
a. Given any
b.
"/_.
real solutions.
5. The reciprocal ofany positive real number
is positive.
a. Given any positive real number r, the recip-rocal
of
b. For any real number r, if r is
_, then
c. If a real number r _, then
6.
rhen
9. For all equations E, if E is quadratic then
E
Given any real number, there is a real number
that is greater.
a. Given any real number r, there is
s such that s is
b. For
-r_,
e. For all squares J , _.
in lgTg by
a set
Georg
intuitively, aI
1.3
'ise Set 7.2
1. Which of the tbllow'ing
c. IJ:lre Zl2<r<-2]
d.V:{seZls>2ors<3}
e.W:{teZll<r<-3}
sets are equal?
B -- {d,e,a,c)
A={a,b,c,dl
D: {a,a,d,e,c,e}
C =ld,b,a,cl
l' *nr. in words how to read each of the following out loud'
,.
lxeR-10<r<l)>
-a.f. ix
l)
e Rlx < Oorx
f. X =
8.
go*
.. ffo*
U.
numtimes
inB
:"
.
many elements are in the set {3' 4' 3' 5}?
many elements are in the set { 1, { I }, {1' { 1i}}?
2' 2l'!
U. How many elements are in the ser 12,2'
c. How many elements are in the set {0' {0)}?
: first
,rtical
:with
16 a.
d. Is {0} e {{0}' {li}?
e. Is 0 e {{0}, {l}}?
'
-
{r e Zl n - (-
Answers
-
1)(, tbr some integer &}.
1
+
{tl, gl.
((-2)r, -2t) = (-22. (-D2)',!
(-l)i, for some integer
Is (8
5)?
e,
AxB
c.AxA
of the following sets.
b. T = {m e Zlm
Is
a.
6. Fo, each integer n,let T, - ln, n?1. How many elements
,,' ue in each of Tz, T-t' Tt and To? Justify your answers'
. 7. Use the set-roster notation to indicate the elements in each
-a. S
: lf, jl, and C :
notation to write each of the following sets, and indicate
the number of elements that are in each set:
tFr
:
B
> l}
- irJ : (-1, -l)?
a rs (_02. (-2)r) : (*, -t),
ll. Let A: lw,x, -r, z) and B : {4, b}. Use the set-roster
c.
A: {0, 1,2}
B-(-reRl-lSx<3I
C={x€Rl-l <x<3}
D-lxeZl -l<x<3)
g-lxez+l-I<x<3)
''
f, ll,
b. Is (5, -5) = (-5,
H 5. Which of the following sets are equal?
:rs. If
, then
lc,d,
< 4orr.r
IsBCA? b. IsCcA?
Is C c C? d. Is C a proper subset of A?
9. a. Is 3 e U,2,31? b. Is I c {1}?
c. Is {2} e {1,2}? d' Is {3i e {1, t2}, {3}i?
f. Is i2} q {1, {2}, t3}i'r
e. Is I e {l}?
c
g. Is {1} {1,2}? h. Is 1 e {{l},2}?
i. Is {l} c {1, t2}}? j. Is {1} c {l}?
4. a. Is 2 e l2l?
=
:
Let A:
<Zlu
a.
c.
1 3. a. Is4={4}?
-
',i
{u
Answer each of the following questions. Give reasons
for your answers.
.. i, e Zlnisafactorof 6)
a. i, eZ+lnisafactorof6)
'
The Language of Relations and Functions 13
12.Let
S:{2,4,6}
b.BxA
d.BxB
and
T:{1,3,5}.
Use the set-roster
notation to write each of the following sets, and indicate
the number of elements that are in each set:
a.,Sx7"
c. SxS
b. IxS
d.TxT
i}.
for Test Yourself
matter 2. the set of all real numbers 3. the set of all integers 4. the set of all rational numbers 5. the set of all x
ttratP(.r) 6.everyelementinAisanelementinB T.thesetof allorderedpairs(a.D)whereaisinAandbisinB
l.
does not
1.3
The Language
Mathematics is a language.
-
of
such
Relations and Functions
Josiah Willard Gibbs (1839-1903)
There are many kinds of relationships in the world. For instance, we say that two people
are related by blood if they share a common ancestor and that they are related by marriage
if one shares a common ancestor with the spouse of the other. We also speak of the relationship between student and teacher, between people who work for the same employer,
and between people who share a common ethnic background'
Similarly, the objects of mathematics may be related in various ways. A set A may
be said to be related to a set B if A is a subset of B, or if A is not a subset of B, or if A
and B have at least one element in common. A number.r may be said to be related to a
number-yifx<y,orifxisafactorofy,orif x2+y2:l.Twoidentifiersinacomputer