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 be can alphabet English in the vowels all of set The 1 Example ou i e a 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 , Jersey New Fred a {1,3,5 set as same the is ,5,5,5} {1,3,3,3,5 Set 4 Example
Transcript
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
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B.A as denoted
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7 Example
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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).
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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
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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
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English theof letters theofset theisset universal theand Let
17 Example
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}{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,
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. 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
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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.
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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.
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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,
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• 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?
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ence.correspond one-to-one isit since invertible is :1 y(y)fy- xxy
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12 Example2 fxf(x)ZZf
.invertiblenot is Therefore, one.-to-onenot is ,111Since
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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
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1
2
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1 where, sequence heConsider t
1 Example
4321 ,...},, {,...},a,a,a{a
/n. a}{a nn
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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
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lower its with starting integers all through runssummation ofindex The
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ummationindex of s
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5 Example
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:100
1j
Solution
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