Topics in Mathematics
Chapter 2 Homework Assignments=
Section 2.1 &=
nbsp; 6,
7, 8, 9, 12, 13-33 odds, 43-49 odds, 67-85 odds
Section 2.2 &=
nbsp; 3,
7-57 odds
Section 2.3 &=
nbsp; 1-14
all, 15, 16, 18, 33-81 odds, 85
Section 2.4 &=
nbsp; 1,
7, 9, 11, 13, 15, 29-60 all, 61-77 odds
Section 2.5 &=
nbsp; 1-15
odds
2.1 Set Theory:
&nbs=
p;
&nbs=
p;
&nbs=
p;
Set:
&nbs=
p;
&nbs=
p;
Elements:
&nbs=
p;
&nbs=
p;
Well-Defined Set:
&nbs=
p;
&nbs=
p;
Two Methods to Describ=
e a
Set
&nbs=
p;
Roster Form:
&nbs=
p;
&nbs=
p;
Ellipsis:
&nbs=
p;
&nbs=
p;
&nbs=
p;
Inclusive:<=
b>
&nbs=
p;
Examples: Write the following set in Roster
Notation.
&nbs=
p;
Set of Na=
tural
Numbers =
&nb=
sp; =
&nb=
sp; =
&nbs=
p;
Set of Od=
d Natural
Numbers =
&nb=
sp; =
&nb=
sp;
Set of na=
tural
numbers between 1 and 5 &n=
bsp; =
; &n=
bsp;
&nbs=
p;
Set of na=
tural
numbers between 1 and 5, inclusive &nb=
sp; =
&nbs=
p;
Set of al=
l the
letters of the alphabet &n=
bsp; =
; &n=
bsp; =
;
&nbs=
p;
Set of the
letters a, b, and c =
&nb=
sp; =
&nb=
sp; =
&nbs=
p;
Set of the
letters between d and h &n=
bsp; =
; &n=
bsp; =
;
&nbs=
p;
Set of the
letters between f and x, inclusive &nb=
sp; =
&nb=
sp;
Set-Builder Notation:<=
o:p>
&nbs=
p;
&nbs=
p;
Examples: Read the following sets in set-bui=
lder
notation. Then express the se=
ts in
roster notation.
&nbs=
p;
A =3D { x=
| x
ε N and x > 5 } &n=
bsp; =
; &n=
bsp; =
;
&nbs=
p;
&nbs=
p;
B =3D { x=
| x
ε N and x < 5 } &n=
bsp; =
; &n=
bsp; =
;
&nbs=
p;
&nbs=
p;
C =3D { x=
| x
ε N and 3 ≤ x < 10 } &nb=
sp; =
&nb=
sp;
&nbs=
p;
&nbs=
p;
D =3D { x=
| x
ε N, x is odd, and x > 100} &n=
bsp;
&nbs=
p;
&nbs=
p;
Examples: Convert the following sets, given =
in
roster notation, to set-builder notation.&=
nbsp;
Then read the sets in set-builder notation.
&nbs=
p;
E =3D {-2=
,-1,0,1,2} &=
nbsp; &nbs=
p; &=
nbsp;
&nbs=
p;
&nbs=
p;
F =3D {11=
,12,13,…,25} &=
nbsp; &nbs=
p; &=
nbsp;
&nbs=
p;
&nbs=
p;
G =3D {10=
,12,14,16,18,20,…} &=
nbsp; &nbs=
p;
&nbs=
p;
&nbs=
p;
H =3D
{1,2,3,…} &nbs=
p; &=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p;
&nbs=
p;
&nbs=
p;
Equality of Sets:
&nbs=
p;
&nbs=
p;
&nbs=
p;
Example:=
&nbs=
p;
Cardinal Number of a S=
et:
&nbs=
p;
Examples: Find the cardinalities of the foll=
owing
sets.
&nbs=
p;
Set of na=
tural
numbers between 1 and 5 &n=
bsp; =
; &n=
bsp;
&nbs=
p;
&nbs=
p;
Set of na=
tural
numbers between 1 and 5, inclusive &nb=
sp; =
&nbs=
p;
&nbs=
p;
Set of al=
l the
letters of the alphabet &n=
bsp; =
; &n=
bsp; =
;
&nbs=
p;
&nbs=
p;
Set of the
letters a, b, and c =
&nb=
sp; =
&nb=
sp; =
&nbs=
p;
&nbs=
p;
Set of the
letters between d and h &n=
bsp; =
; &n=
bsp; =
;
&nbs=
p;
&nbs=
p;
Set of the
letters between f and x, inclusive &nb=
sp; =
&nb=
sp;
&nbs=
p;
&nbs=
p;
Equivalency of Sets:
&nbs=
p;
&nbs=
p;
&nbs=
p;
Example:<=
span
style=3D'mso-spacerun:yes'> Determine if the following sets are
equivalent.
&nbs=
p;
A =3D {re=
d, blue,
yellow, green} =
; &n=
bsp;
B =3D {1,=
2,3,4}
&nbs=
p;
&nbs=
p;
C =3D {Pi=
rates,
Steelers, Penguins} =
&nb=
sp; =
D =3D {ho=
ckey,
baseball, football}
&nbs=
p;
&nbs=
p;
E =3D {France, En=
gland,
Germany} &=
nbsp; &nbs=
p; &=
nbsp;
F =3D {London, Rome, Berlin, Paris}
Finite Set:=
&nbs=
p;
&nbs=
p;
&nbs=
p;
Null Set (Empty Set):<=
/span>
&nbs=
p;
&nbs=
p;
Universal Set:<=
/b>
&nbs=
p;
&nbs=
p;
Examples:=
&nbs=
p;
Indicate =
the set
of states that begin with Z. &nb=
sp; =
&nb=
sp;
&nbs=
p;
&nbs=
p;
What is t=
he universal
set for the above question? &nbs=
p; &=
nbsp;
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
2.2 Subsets:
&nbs=
p;
&nbs=
p;
&nbs=
p;
Examples: Determine whether set A is a subse=
t of
set B. If it is not a subset,=
prove
such (find an element in set A that is not in set B).
&nbs=
p;
A =3D
{10,12,14,…} &=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p;
B =3D
{2,4,6,…}
&nbs=
p;
&nbs=
p;
A =3D {x =
| x ε
N and x is odd} &nbs=
p; &=
nbsp; &nbs=
p;
B =3D {x =
| x ε
N and x > 10}
&nbs=
p;
&nbs=
p;
A =3D
{1,2,3,…,10} &=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p; =
B =3D {x =
| x ε
N and 0 < x < 11}
&nbs=
p;
&nbs=
p;
A =3D { } &nbs=
p; &=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p;
B =3D {3,=
4,9} &=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p; &=
nbsp;
&nbs=
p;
&nbs=
p;
Proper Subset:<=
/b>
&nbs=
p;
&nbs=
p;
&nbs=
p;
Examples: Determine whether set A is a proper
subset of set B.
&nbs=
p;
A =3D {4,=
6,8,…} &=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p;
B =3D {2,=
4,6,…}
&nbs=
p;
&nbs=
p;
A =3D {1,=
2,3,4,5} &=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p; =
B =3D {5,=
4,3,2,1}
&nbs=
p;
&nbs=
p;
A =3D {x =
| x =3D 10} &=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p; =
B =3D {x | x ≥ 10}
Differences
Between an “Element of” and a “Subset of”
For ☺ to be an E=
lement
of a Set:
☺ ![](3D"chapter_2_skeleton_notes_files/image002.gif")
{ ☺ }
For ☺ to be a Su=
bset of
a Set:
{☺} ![](3D"chapter_2_skeleton_notes_files/image004.gif")
{ ☺ }
Examples: Determine if the following are Tru=
e or
False.
&nbs=
p;
1
{1,2,3} &=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p; &=
nbsp;
{1}
{1,2,3} &=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p; &=
nbsp;
1
{{1},{2},{3=
}} &=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p;
{1}
{{1},{2},{3=
}} &=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p;
1
{1,2,3} &=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p; &=
nbsp;
{1}
{1,2,3} &=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p; &=
nbsp;
1
{{1},{2},{3=
}} &=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p;
{1}
{{1},{2},{3=
}} &=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p;
{{1}} {{1}.{2},{3=
}} &=
nbsp; &nbs=
p; &=
nbsp;
Number of Subsets=
&nbs=
p;
Consider:
|
Set
|
Subsets
|
Number of Subsets
|
|
{ }
|
|
|
|
{a}
|
|
|
|
{a,b}
|
|
|
&nbs=
p;
&nbs=
p;
Number of Distinct Sub=
sets
of a Finite Set =3D
&nbs=
p;
&nbs=
p;
&nbs=
p;
Example: Determine the number of distinct s=
ubsets
for the set {a,b,c}. List
them. How many are proper sub=
sets?
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
Example: Determine the number of distinct s=
ubsets
for the set {w,x,y,z}. List
them. How many are proper sub=
sets?
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
2.3 Venn Diagrams, Set Operations=
&nbs=
p;
Venn Diagram:
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
Example: Set A in a Venn Diagram.
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
Example: Set A and Set B in a Ve=
nn
Diagram. There are four
possibilities.
&nbs=
p;
Case 1: Disjoint sets
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
Case 2: Subsets
&nbs=
p;
&nbs=
p;
&nbs=
p;
Case 3: Equal sets
&nbs=
p;
&nbs=
p;
&nbs=
p;
Case 4: Overlapping Sets=
&nbs=
p;
&nbs=
p;
Standard Two Set Venn
Diagram:
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
Two Set Venn Diagram
with Regions Labeled
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
The Four Cases with a
Standard Venn Diagram
&nbs=
p;
Let U =3D
{1,2,3,4,5}
&nbs=
p;
![](3D"chapter_2_skeleton_notes_files/image006.gif")
Case 1: Disjoint Sets =
&nb=
sp; =
=
A =3D {1,=
2}
B =3D {3,=
4}
&nbs=
p;
&nbs=
p;
Case 2: Subsets &nb=
sp; =
&nb=
sp; =
A =3D {2,=
3}
B =3D {2,=
3,4}
&nbs=
p;
&nbs=
p;
Case 3: Equal sets =
&nb=
sp; =
A =3D {3,=
4,5}
B =3D {3,=
4,5}
&nbs=
p;
&nbs=
p;
Case 4: Overlapping Sets =
&nb=
sp; =
A =3D {2,=
3}
B =3D {3,=
4,5}
&nbs=
p;
&nbs=
p;
&nbs=
p;
Set Operations:=
&nbs=
p;
&nbs=
p;
&nbs=
p;
Complement:=
&nbs=
p;
&nbs=
p;
Examples: Given U =3D {a,b,c,d,e,f}, A =3D {=
a,b,c}, B
=3D {b,c,d,e}, and C =3D { },
find:
&nbs=
p;
&nbs=
p;
A’ &=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p; &=
nbsp;
B’ &=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p; &=
nbsp;
C’ &=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p; &=
nbsp;
&nbs=
p;
&nbs=
p;
Intersection:
&nbs=
p;
&nbs=
p;
&nbs=
p;
Examples: Given U =3D {a,b,c,d,e,f}, A =3D {=
a,b,c}, B
=3D {b,c,d,e}, and C =3D { }, find:
&nbs=
p;
&nbs=
p;
A
![](3D"chapter_2_skeleton_notes_files/image011.gif")
B &=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p; &=
nbsp;
A
C &=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p; &=
nbsp;
A’
B &=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p; &=
nbsp;
(A
B)’ &=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p;
A
U &=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p; &=
nbsp;
&nbs=
p;
&nbs=
p;
Union:
&nbs=
p;
&nbs=
p;
&nbs=
p;
Examples: Given U =3D {a,b,c,d,e,f}, A =3D {=
a,b,c}, B
=3D {b,c,d,e}, and C =3D { },
find:
&nbs=
p;
&nbs=
p;
A
![](3D"chapter_2_skeleton_notes_files/image014.gif")
B &=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p; &=
nbsp;
A
C &=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p; &=
nbsp;
A’
B &=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p; &=
nbsp;
(A
B)’ &=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p;
A
U &=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p; &=
nbsp;
&nbs=
p;
&nbs=
p;
Relationship of n(A B), n(A), n=
(B),
and n(A B)
&nbs=
p;
Consider =
A =3D
{a,b,c} and B =3D {b,c,d,e}:
&nbs=
p;
n(A) =3D =
3 &=
nbsp; n(B)
=3D 4 &n=
bsp; What
is n(A B)?
&nbs=
p;
&nbs=
p;
&nbs=
p;
For any finite sets A =
and B:
&nbs=
p;
&nbs=
p;
&nbs=
p;
Examples: Set C has 10 elements, set D has 14
elements, and there are 6 elements common to the two sets. Find the number of elements in (C =
D).<=
/p>
&nbs=
p;
&nbs=
p;
&nbs=
p;
2.4 Venn Diagrams with 3 S=
ets:
&nbs=
p;
&nbs=
p;
Set Operations &=
nbsp; &nbs=
p; Represent
Region(s)
A ∩ B ∩ C
A ∩ B
A ∩ C =
&nb=
sp; =
&nb=
sp;
B ∩ C =
&nb=
sp; =
&nb=
sp;
A U B =
&nb=
sp; =
&nb=
sp;
A U C =
&nb=
sp; =
&nb=
sp;
B U C =
&nb=
sp; =
&nb=
sp;
A U B U C =
&nb=
sp; =
&nb=
sp;
(A U B U C)’ &=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p;
Constructing a Venn Di=
agram:
&nbs=
p;
Example: Create a Venn Diagrams using the
following set.
&nbs=
p;
U =3D {1,=
2,3,4,5,6,7,8,9,10}
A =3D {2,=
4,6,8,10}
B =3D {2,=
5,8}
C =3D
{5,6,7,8,9,10}
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
<=
/o:p>
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
Example: Create a Venn Diagram using the
following set.
&nbs=
p;
U =3D
{a,b,c,d,e,f,g,h,i,j,k,l}
A =3D {a,=
b,c}
B =3D {a,=
c,e,g,i,k}
C =3D {a,=
d,g,j}
&nbs=
p;
&nbs=
p;
&nbs=
p;
<=
/o:p>
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
Equality of Sets:
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
&nbs=
p;
Example: Determine whether (A ∩ B)=
217; =3D
A’ U B’ for all sets.
&nbs=
p;
|
(A ∩ B)’=
|
A’ U B’<=
/span>
|
|
Set
|
Regions
|
Set
|
Regions
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
&nbs=
p;
Example: Determine whether A U (B ∩ C=
) =3D (A
U B) ∩ (A U C) for all sets.
&nbs=
p;
|
A =
U (B
∩ C)<=
/p>
|
(A=
U B)
∩ (A U C)
|
|
Set
|
Regions
|
Set
|
Regions
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
&nbs=
p;
De Morgan’s Laws=
: For all sets A and B,
&nbs=
p;
Example: We have verified Law 2. Verify Law 1 is true for all sets.=
&nbs=
p;
|
(A U B)’
|
A’ ∩ B&#=
8217;
|
|
Set
|
Regions
|
Set
|
Regions
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
&nbs=
p;
2.5<=
span
style=3D'mso-spacerun:yes'> Applications with Sets<=
/span>
&nbs=
p;
Country-W=
estern
songs emphasize three basic themes: love, prison, and trucks. A survey of the songs played freque=
ntly
by a local Country-Western radio station produced the following data:
2 songs =
about
truck drivers who are in love while in prison
10 songs=
about
prisoners in love
11 songs=
about
truck drivers in love
7 songs =
about truck
drivers in prison
23 songs=
about
people in love
22 songs=
about
truck drivers
22 songs=
about
prisoners
3 songs =
about
people who are not in prison, are not in love, and do not drive trucks
&nb=
sp;
(a) How many songs were surveyed?
(b) How many songs were about truck dr=
ivers
who are not in prison?
(c) How many songs were about people in
prison who are not in love and do not drive trucks?
A farmer has 36 pigs.=
A survey of these pigs produced the following data:
16 healthy pigs
18 fat pigs
18 greedy pigs
11 fat pigs that are also =
greedy
8 healthy fat pigs
4 pigs that are healthy an=
d greedy
3 pigs that are fat, greed=
y, and
healthy
(a) How many pigs have none of the thr=
ee
attributes?
(b) How many pigs are greedy but neith=
er fat
nor healthy?
(c) How many pigs are fat and healthy =
but
not greedy?
(d) How many pigs possess exactly two =
of the
three attributes?
&=
nbsp;
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