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1.4 Sets
Definition 1. A set is a group of objects . The objects in a set are called the elements, or members, of the set.
.,,,V as written becan alphabet English in the vowelsall ofset The
1 Example
ouiea
Example 2
The set of positive integers less than 100 can be denoted as .99,...,3,2,1
Definition 2. Two sets are equal if and only if they have the same elements.
Example 3
A set can also consists of seemingly unrelated elements: . ,,2, JerseyNewFreda
{1,3,5}.set as same theis ,5,5,5}{1,3,3,3,5Set
4 Example
• A set can be described by using a set builder notation.
}10 than lessinteger positive oddan is{
4 Example
x | x O
numbers} real|{R
}{1,2,3,...integers} positive|{Z
}1012{integers} |{Z
..}{0,1,2,3,.numbers} natural|{N
5 Example
x
x
,...,,,-...,-x
x
• A set can be described by using a Venn diagram.
Example 6
Draw a Venn diagram that presents V, the set of vowels in English alphabet.
a,e,i,o,uV
U
Definition 3. The set A is said to be a subset of B if and only if every element of A is also an element of B. We use the notation to indicate that A is a subset of the set B.BA
• The set that has no elements is called empty set, denoted by .
. then , and If
. and ,set any For
ifonly and if
BAABBA
PPPP
B)xAx(xBA
B.A as denoted
B, ofsubset proper a be tocalled isA B,Abut that Bset ofsubset a isA set a If
A B
U
b}}.{a,set theofsubset a is { b}}{a,{b},{a},,{
7 Example
xx|
Definition 4. Let S be a set. If there are exactly n distinct elements in S , where n is a nonnegative integer, we say that S is a finite set and n is the cardinality of S. The cardinality of S is denoted by |S|.
.26Then alphabet.English in the letters ofset thebe SLet
8 Example
|S|
elements. no hasset empty thesince 0,| |
9 Example
Definition 5. A set is said to be infinite if it is not finite.
Example 10
The set of positive integers is infinite.
The Power Set
Definition 5. The power set of a set S is the set of all subsets of S, denoted by P(S).
}}.2,1,0{},2,1{},2,0{},1,0{},2{},1{},0{,{})2,1,0({:
{0,1,2}?set theofset power theisWhat
11 Example
PSolution
Cartesian Products
element.th its as and .,element,.. second its as
element,first its as has that collection ordered theis ),..., tuple-order The
7. Definition
2
121
naa
aa,a (an
n
n
pairs. ordered called are tuples-2
.,...,2,1for ifonly and if ),...,,(),...,,( 2121
nibabbbaaa iinn
. Hence, . and where, paris ordered all
ofset theis ,by denoted , and ofproduct Cartisian The sets. be B andA Let
8Definition
B}bA{(a,b)| aBABbAa(a,b)
BABA
ABBA
)}),(c),(c),(b),(b),(a{(aAB
,c)},b),(,a),(,c),(,b),(,a),({(BASolution
ABBA{a,b,c}B},{A
2,1,2,1,2,1,
222111 :
? and productsCartesian theis What . and 21Let
12 Example
.21for ,...,
sother wordIn .21for where,,..., tuple- ordered
ofset theis ,by denoted ,,...,, ofproduct Cartisian The
8Definition
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2121
,...,n},iA)| aa,a{(aAAA
,...,n,iAa)a,a(an
AAAAAA
inn
in
nn
)},,),(,,),(,,),(,, (
),,,),(,,),(,,),(,,),(,,),(,,),(,,),(,,(CBASolution
CBCBA
221121021211
111011220120020210110010{:
?}2,1,0{ and }2,1{},1,0{A where,product Cartesian theisWhat
13 Example
1.5 Set Operations
BADefinition 1. Let A and B be sets. The union of the sets A and B, denoted by , is the set that contains those elements that are either A and B, or in both. That is }.|{ BxAxxBA
Definition 2. Let A and B be sets. The intersection of the sets A and B, denoted by , is the set that contains those elements that are in both A and B. That is }.|{ BxAxxBA
BA
}.3,1{}3,2,1{}5,3,1{ }.5,3,2,1{}3,2,1{}5,3,1{
14 Example
A B
U
B
U
Definition 3. Two sets are called disjoint if their intersection is the empty set.
.B A
φ,BA},,,,{B} ,,,,{A
disjoint are and
Since .108642 and97531Let
15 Example
. have wesets, ofunion theofy cardinalit For the B||B|-|A|A|B||A
Definition 4. Let A and B be sets. The difference of A and B, denoted by A-B is the set containing of those elements that are in A but not in B. The difference of A and B is also called the complement of B with respect to A. That is,
}.|{ BxAxxBA
A B
U
{5}{1,2,3}-{1,3,5}
16 Example
Definition 5. Let U be the universal set. The complement of the set A, denoted by , is the complement of A with respect to U. In other words, the complement of the set is U-A. That is,
A}.|{ AxxA
A
U
A
}.,,,,,,,,,,,,,,,,,,,,{hen alphabet.T
English theof letters theofset theisset universal theand Let
17 Example
zyxwvtsrqpnmlkjhgfdcbA
}{a,e,i,o,uA
Set Identities
Table 1 Set Identities Identity Name
Identity laws
Domination laws
Idempotent laws
Complementation laws
Commutative laws
Associative laws
Distributive laws
De Morgan’s law
AUA
AA
A
UUA
AAA
AAA
AA )(
ABBA
ABBA
CBACBA
CBACBA
)()(
)()(
)()()(
)()()(
CABACBA
CABACBA
BABA
BABA
• One way to prove that two sets are equal is to show that one of sets is a subset of the other and vise versa.
other. theofsubset a isset each that showingby that Prove
18 Example
BABA
B.ABABAxBAx
BA or xxBxAxBAx
BABAx
BA or xxBAxBAx
Solution
that shows Tis . Thenfore, .Hence,
. that followsIt .or then if Next,
.BA that shows This . Hence,
. that implies This . then if First,
:
• One way to prove that two sets are equal is to use set builder and the rules of logic.
.BA that show toesequivalenc logical andbuilder set Use
19 Example
BA
}.{}|{
}|{)}(|{
))}((|{}|{
:
BAxBxAxx
BxAxxBxAxx
BAxxBAxxBA
Solution
• Set identities can be proved by using membership tables.
C).(AB)(AC)(BA that show to tablemembership a Use
20 Example
Table 2. A membership table for the distributive Property
A B C CB )( CBA BA CA )()( CABA
1 1 11 1 01 0 11 0 00 1 10 1 00 0 10 0 0
11101110
11100000
11000000
10100000
11100000
• Set identities can be established by those that we have already proved.
.
that Show
21 Example
A)BC(C)(BA
unions.for law ecommutativ by the )(
onsintersectifor law ecommutativ by the )(
law sMorgan' De second by the )(
law sMorgan' Defirst by the )()(
:
ABC
ACB
CBA
CBACBA
Solution
1.6 Functions
Definition 1. Let A and B be sets. A function f from A to B is an assignment of exactly one element of B to each element of A. We write f(a)=b if b is the unique element of B assigned by the function f to the element a of A. If f is a function from A to B, we write f : A B.
Example 1
Let set A={Adams, Chou, Goodfriend, Rodriguez, Stevens} and B={A,B,C,D,F}.
Let G be the function that assigns a grade to a student in our discrete mathematics.
Adames
Chou
Goodfriend
Rodriguez
Stevens
A
B
C
D
F
G
The domain of G is the set A={Adams, Chou, Goodfriend, Rodriguez, Stevens}, and the range of G is the set {A,B,C,F}.
xy
z
A function
x
Not a function
Definition 2. If f is a function from A to B, we say that A is the domain of f and B is the codomain of f. If f(a)=b, we say that b is the image of a and a is a pre-image of b. The range of f is the set of all images of elements of A. Also, if f is a function from A to B, we say that f maps A to B.
f
a b=f(a)
A B
f
Z. to Zfrom one theis xf(x)Function
2 Example2
The domain and codomain of f is Z, and the range of f is the set {0,1,4,9,…}.
(x).(x)ff)(x)f (f
(x),f(x)f)(x)f (f
RA f fff
RA ff
2121
2121
2121
21
by defined to from functions also areand Then
. to fromfunction thebe and Let 3. Definition
? and functions theis What .)(
and such that to fromfunction thebe Let
3 Example
21212
2
2121
ffffxxxf
x fRRf and f
.)()()())((
,)()()())((
:
43222121
222121
xxxxxxfxfxff
xxxxxfxfxff
Solution
Definition 4. Let f be a function from the set A to the set B and let S be a subset of A. The image of S is the subset of B that consists of the images of elements of S. We denote the image of S by f(S), so that
}.|)({)( SssfSf
S f(S)A
B
Example 4
Let A={a,b,c,d,e} and b={1,2,3,4} with f(a)=2,
f(b)=1,f(c )=4, f(d)=1, and f(e)=1. The image of
S={b,c,d} is the set f(S)={1,4}.
One-to-One and Onto Functions
Definition 5. A function is said to be one-to-one, or injective, if and only if f(x)=f(y) implies that x=y for all x and y in the domain of f. A function is said to be an injection if it is one-to-one.
x
y
f(x)f(y)
function but not one-to-one
x
yf(x) f(y)
one-to-one function
Example 6
Determine whether the function f from {a,b,c,d}
to {1,2,3,4,5} with f(a)=4, f(b)=5, f(c )=1, f(d)=3
is one to one. a
b
c
d
12
34
5
one?-to-one integers
ofset the tointegers ofset the
from xf(x)function theIs
7 Example2
.111 instance,for
because, one-to-onenot isIt
:
)f(-)f(
Solution
Definition 6. A function f whose domain and codomain are subsets of the set of real numbers is called strictly increasing if f(x)<f(y) whenever x<y and x and y are in the domain of f. Similarly, f is called strictly decreasing if f(x)>f(y) whenever x<y and x and y are in the domain of f.
• A strictly increasing or strictly decreasing function must be one-to-one.
onto. isit if surjection a called isfunctionA
withelement an is thereelement every for ifonly and if
,surjectiveor onto, iscalled to from function A 7. Definition
f
b. f(a)AaBb
BAf
Example 8
Determine whether the function f from {a,b,c,d}
to {1,2,3} with f(a)=3, f(b)=2, f(c )=1, f(d)=3
is onto. a
b
c
d
1
2
3
BA
onto
A
into
B
onto? integers ofset the tointegers ofset thefrom function theIs
9 Example2xf(x)
instance.for ,1 withinteger no is theresincenot, isIt
:2 xx
Solution
Definition 8 . The function f is a one-to-one correspondence, or a bijection, if it is both one-to-one and onto.
x
yf(x) f(y)
one-to-one
A B A B
onto
+
Example 10
one-to-one, not onto
a
b
c
123
4
onto, not one-to-one
a
b
c
12
3
a 1
one-to-one and onto
b
c
23
4e
1
neither one-to-one nor onto
a
b
c
23
4e
not a function
a
b
c
123
4
Inverse Function and Compositions of fuctions
.)( when )(f Hence,
.by denoted is offunction inverse The .such that in element unique the
tobelonging elementan toassignshat function t theis offunction inverse The
.set toset thefrom encecorrespond one-to-one a be Let 9. Definition
1-
1
bafab
ff bf(a)Aa
B bf
BAf
-
)(1 bfa
1f
f
1f
f )(afb
A B
• A function is invertible if it is one-to-one correspondence, and it is not invertible if it is not one-to-one correspondence.
Example 11
Let f be the function from the set of integers to the set of integers such that f(x)=x+1. Is f invertible, and if it is, what is its inverse?
.1 Therefore, .1 then, of image theis that Suppose
ence.correspond one-to-one isit since invertible is :1 y(y)fy- xxy
fSolution-
?invertible Is . with to fromfunction thebe Let
12 Example2 fxf(x)ZZf
.invertiblenot is Therefore, one.-to-onenot is ,111Since
:
f f)f() f(-
Solution
aA
g(a)B
f(g(a))Cg f
g f
gf
gf Example 13
Let f and g be the functions from the set of integers to the set of integers defined by f(x)=x+3 and g(x)=3x+2. What are ? and nscompositio the fggf
116)32())(())((
76)23())(())((
:
xxgxfgxfg
xxfxgfxgf
Solution
Definition 10 . Let g be a function from set A to the set B and let f be a function from the set B to set C. The composition of the functions f and g, denoted by f g, is defined by )).(())(( agfagf
• Let f be a one-to-one correspondence function from set A to set B and
be the inverse of f. 1f
.)())(())(( and ,)())(())((
Hence, .then ,) if and ;) then , If11111
11
bafaffaffabfaffaff
bf(a)a(bfa(bfbf(a) --
Some Important Functions
777 ,41.3 ,33.1 ,12
1 ,0
2
1
14. Example
nn xaxaxaa 2
210
Other Functions• Polynomial functions • logarithmic functions• exponential functions
)2 when log( log bxxb
)e ,2( xxxa
.x xx
x
xxx
xtionfloor func
by denoted is at function ceiling theof valueThe . toequalor than
greater isat integer thsmallest thenumber real theassign tofunction ceiling The
.by denoted is at function floor theof valueThe . toequalor than less isthat
integer largest thenumber real the toassigns The 12. Definition
1.7 Sequences and Summations
Definition 1. A sequence a is function from a subset of the set of integers to a set S. We use the notation to denote the image of the integer n. We call a term of the sequence.
na na
sequence. thedescribe to}{a use We n
1 n2 1a 2a na
.3
1
2
11 withbegins sequence theoflist The
1 where, sequence heConsider t
1 Example
4321 ,...},, {,...},a,a,a{a
/n. a}{a nn
.1111 with begins sequence theoflist The
.1 where, sequence heConsider t
2 Example
3210 ,...},,,{,...},b,b,b{b
)(b}{b nnn
Special Integer Sequences
Finding a formula or a general rule for constructing the terms of a sequence.
• Are there are runs of the same value?
• Are terms obtained from previous terms by adding or multiplying a particular amount?
• Are the terms obtained by combining previous terms in a certain way?
Example 3.
What is a rule that can produce the terms of a sequence if the first 10 terms are 1,2,2,3,3,3,4,4,4,4?
Example 4.
What is a rule that can produce the terms of a sequence if the first 10 terms are 5,11,17,23,29,35,41,47,53,59?
Solution:
A reasonable guess is that the nth term is 5+6(n-1)=6n-1.
Summations
.limit upper its with ending and limit
lower its with starting integers all through runssummation ofindex The
aaa
is,That le.any variab be could which thecalled is
letter theHere .represent toanotation use We
n
mkk
n
mii
n
mjj
1m
n
mjj
nm
ummationindex of s
jaaa nm
,....321for 1
where
, sequence theof terms100first theof sum theExpress
5 Example
,,nn
a
}{a
n
n
. j
1 is sum The
:100
1j
Solution
?j of value theisWhat
6 Example5
1j
2
5554321
:
222225
1
2 j
j
Solution
follows. as compute We. called are
sumssuch arise;commonly nsprogressio geometric of termsof Sums
numbers. real are , the, and term, the, where
,
form theof sequence a is A
7 Example
0
32
n
j
j
k
arSseriesgeometric
ratiocommonrinitiala
,...,ar,ar a, ar, ar
on progressigeometric
.1 then ,1if ,1
S then 1r if Therefore,
)()(
)(
1
11
0
11
10
1
0
)a(nS rr
aar
aarSaarS
aarararararrrS
n
nn
n
k
nkn
k
kn
j
jn
j
j
4
1i
3
1j
. sum double theEvaluate
8 Example
ij
6024181266)3(
:4
1i
4
1i
4
1i
3
1j
iiiiij
Solution
4
)1(
6
)12)(1(2
)1(
1,1
Formulae.
Summation UsefulSome 1 Table
22
1
3
1
2
1
1
0
nnk
nnnk
nnk
rr
aarar
n
k
n
k
n
k
nn
k
k
100
50k
2.k Find
9 Example
.297925 6
201101100
6
201101100
kkk
:49
1k
2100
1k
2100
50k
2
Solution