2. Sets
DEF: A set is a collection of elements.
This is another example where mathematics
must start at the level of intuition. Sets are the
basic data structure out of which most
mathematical theories are built. For many
years mathematicians hoped that sets could be
defined directly from logic, thus giving a full-
proof foundation to Mathematics, when
compared to other sciences. Effort failed!
2
3. Sets
Curly braces ―{― and ―}‖ are used to denote
sets.
Java note: In Java curly braces denote arrays,
a data-structure with inherent ordering.
Mathematical sets are unordered so different
from Java arrays. Java arrays require that all
elements be of the same type. Mathematical
sets don’t require this, however. EG:
{ 11, 12, 13 }
{ , , }
{ , , , 11, Leo }
3
4. Sets
A set is defined only by the elements
which it contains. Thus repeating an
element, or changing the ordering of
elements in the description of the set,
does not change the set itself:
{ 11, 11, 11, 12, 13 } = { 11, 12, 13 }
{ , , }={ , , }
4
5. Standard Numerical Sets
The natural numbers:
N = { 0, 1, 2, 3, 4, … }
The integers:
Z = { … -3, -2, -1, 0, 1, 2, 3, … }
The positive integers:
Z+ = {1, 2, 3, 4, 5, … }
The real numbers: R --contains any decimal number of arbitrary
precision
The rational numbers Q: these are numbers whose decimal expansion
repeats; Q are numbers that can be represented in the form a/b where a
Z and b Z+
Q: Give examples of numbers in R but not Q.
5
7. -Notation
The Greek letter ―‖ (epsilon) is used to denote
that an object is an element of a set. When
crossed out ―‖ denotes that the object is
not an element.‖
EG: 3 S reads:
―3 is an element of the set S ‖.
Q: Which of the following are true:
1. 3R
2. -3 N
3. -3 R
4. 0 Z+
5. x xR x2=-5
7
8. -Notation
A: 1, 3 and 4
1. 3 R. True: 3 is a real number.
2. -3 N. False: natural numbers don’t
contain negatives.
3. -3 R. True: -3 is a real number.
4. 0 Z+. True: 0 isn’t positive.
5. x xR x2=-5 . False: square of a
real number is non-neg., so can’t be -5.
8
9. -Notation
DEF: A set S is said to be a subset of the set T
iff every element of S is also an element of
T. This situation is denoted by
ST
A synonym of ―subset‖ is ―contained by‖.
Definitions are often just a means of
establishing a logical equivalence which aids
in notation. The definition above says that:
ST x (xS ) (xT )
We already had all the necessary concepts, but
the ―‖ notation saves work.
9
10. -Notation
When ―‖ is used instead of ―‖, proper
containment is meant. A subset S of T is
said to be a proper subset if S is not equal
to T. Notationally:
ST S T x (x S xT )
Q: What algebraic symbol is reminiscent of?
10
12. The Empty Set
The empty set is the set containing no
elements. This set is also called the null
set and is denoted by:
{}
12
13. Subset Examples
Q: Which of the following are true:
1. NR
2. ZN
3. -3 R
4. {1,2} Z+
5.
6.
13
14. Subset Examples
A: 1, 4 and 5
1. N R. All natural numbers are real.
2. Z N. Negative numbers aren’t natural.
3. -3 R. Nonsensical. -3 is not a subset but an
element! (This could have made sense if we
viewed -3 as a set –which in principle is the
case– in this case the proposition is false).
4. {1,2} Z+. This actually makes sense. The set
{1,2} is an object in its own right, so could be
an element of some set; however, {1,2} is not
a number, therefore is not an element of Z.
5. . Any set contains itself.
14
6. . No set can contain itself properly.
15. Cardinality
The cardinality of a set is the number of
distinct elements in the set. |S |
denotes the cardinality of S.
Q: Compute each cardinality.
1. |{1, -13, 4, -13, 1}|
2. |{3, {1,2,3,4}, }|
3. |{}|
4. |{ {}, {{}}, {{{}}} }|
15
16. Cardinality
Hint: After eliminating the redundancies just
look at the number of top level commas and
add 1 (except for the empty set).
A:
1. |{1, -13, 4, -13, 1}| = |{1, -13, 4}| = 3
2. |{3, {1,2,3,4}, }| = 3. To see this, set S =
{1,2,3,4}. Compute the cardinality of {3,S, }
3. |{}| = || = 0
4. |{ {}, {{}}, {{{}}} }|
= |{ , {}, {{}}| = 3
16
17. Cardinality
DEF: The set S is said to be finite if its
cardinality is a nonnegative integer.
Otherwise, S is said to be infinite.
EG: N, Z, Z+, R, Q are each infinite.
Note: We’ll see later that not all infinities are
the same. In fact, R will end up having a
bigger infinity-type than N, but surprisingly,
N has same infinity-type as Z, Z+, and Q.
17
18. Power Set
DEF: The power set of S is the set of all
subsets of S.
Denote the power set by P (S ) or by 2s .
The latter weird notation comes from the
following.
Lemma: | 2s | = 2|s|
18
19. Power Set –Example
To understand the previous fact consider
S = {1,2,3}
Enumerate all the subsets of S :
0-element sets: {} 1
1-element sets: {1}, {2}, {3} +3
2-element sets: {1,2}, {1,3}, {2,3} +3
3-element sets: {1,2,3} +1
Therefore: | 2s | = 8 = 23 = 2|s|
19
20. Ordered n-tuples
Notationally, n-tuples look like sets except that
curly braces are replaced by parentheses:
( 11, 12 ) –a 2-tuple aka ordered pair
( , , ) –a 3-tuple
( , , , 11, Leo ) –a 5-tuple
Java: n -tuples are similar to Java arrays ―{…}‖,
except that type-mixing isn’t allowed in Java.
20
21. Ordered n-tuples
As opposed to sets, repetition and
ordering do matter with n-tuples.
(11, 11, 11, 12, 13) ( 11, 12, 13 )
( , , )( , , )
21
22. Cartesian Product
The most famous example of 2-tuples are
points in the Cartesian plane R2. Here ordered
pairs (x,y) of elements of R describe the
coordinates of each point. We can think of the
first coordinate as the value on the x-axis and
the second coordinate as the value on the y-
axis.
DEF: The Cartesian product of two sets A
and B –denoted by A B– is the set of all
ordered pairs (a, b) where aA and bB .
Q: Describe R2 as the Cartesian product of two
sets.
22
23. Cartesian Product
A: R2 = RR. I.e., the Cartesian plane is
formed by taking the Cartesian product of
the x-axis with the y-axis.
One can generalize the Cartesian product
to several sets simultaneously.
Q: If A = {1,2}, B = {3,4}, C = {5,6,7}
what is A B C ?
23
24. Cartesian Product
A: A = {1,2}, B = {3,4}, C = {5,6,7}
A B C =
{ (1,3,5), (1,3,6), (1,3,7),
(1,4,5), (1,4,6), (1,4,7),
(2,3,5), (2,3,6), (2,3,7),
(2,4,5), (2,4,6), (2,4,7) }
Lemma: The cardinality of the Cartesian
product is the product of the cardinalities:
| A1 A2 … An | = |A1||A2| … |An|
Q: What does S equal?
24
25. Cartesian Product
A: From the lemma:
|S | = |||S | = 0|S | = 0
There is only one set with no elements –
the empty set– therefore, S must be
the empty set .
One can also check this directly from the
definition of the Cartesian product.
25
28. Agenda
Set Operations
Union and Disjoint union
Intersection
Difference “-”
Complement “ ”
Symmetric Difference
29. Universe of Reference
When talking about a set, a universe of
reference (universal set ) needs to be
specified. Even though a set is defined
by the elements which it contains, those
elements cannot be arbitrary. If arbitrary
elements are allowed paradoxes can
result arising from self reference.
29
30. Set Builder Notation
Up to now sets have been defined using the curly
brace notation ―{ … }‖ or descriptively ―the set of
all natural numbers‖. The set builder notation
allows for concise definition of new sets. For
example
{ x | x is an even integer }
{ 2x | x is an integer }
are equivalent ways of specifying the set of all
even integers.
35
31. Set Builder Notation
In general, one specifies a set by writing
{ f (x ) | P (x ) }
Where f (x ) is a function of x (okay we haven’t really
gotten to functions yet…) and P (x ) is a propositional
function of x. The notation is read as
―the set of all elements f (x ) such that P (x ) holds‖
Stuff between ―{― and ―|‖
specifies how elements look
Stuff between the ―|‖ and ―}‖
gives properties elements satisfy
Pipe symbol ―|‖ is
short-hand for ―such that‖.
36
32. Set Builder Notation.
Shortcuts.
To specify a subset of a pre-defined set, f (x )
takes the form xS. For example
{x N | y (x = 2y ) }
defines the set of all even natural numbers
(assuming universe of reference Z).
When universe of reference is understood,
don’t need to specify propositional function EG:
{ x 3 | } or simply {x 3 } specifies the set of
perfect cubes
{0,1,8,27,64,125, …}
assuming U is the set of natural numbers.
37
33. Set Builder Notation.
Examples.
Q1: U = N. { x | y (y x ) } = ?
Q2: U = Z. { x | y (y x ) } = ?
Q3: U = Z. { x | y (y R y 2 = x )} = ?
Q4: U = Z. { x | y (y R y 3 = x )} = ?
Q5: U = R. { |x | | x Z } = ?
Q6: U = R. { |x | } = ?
38
34. Set Builder Notation.
Examples.
A1: U = N. { x | y (y x ) } = { 0 }
A2: U = Z. { x | y (y x ) } = { }
A3: U = Z. { x | y (y R y 2 = x )}
= { 0, 1, 2, 3, 4, … } = N
A4: U = Z. { x | y (y R y 3 = x )} = Z
A5: U = R. { |x | | x Z } = N
A6: U = R. { |x | } = non-negative reals.
39
35. Set Theoretic Operations
Set theoretic operations allow us to build new sets
out of old, just as the logical connectives allowed
us to create compound propositions from simpler
propositions. Given sets A and B, the set theoretic
operators are:
Union ()
Intersection ()
Difference (-)
Complement (―—‖)
Symmetric Difference ()
give us new sets AB, AB, A-B, AB, andA .
40
36. Venn Diagrams
Venn diagrams are useful in representing
sets and set operations. Various sets are
represented by circles inside a big
rectangle representing the universe of
reference.
41
37. Union
Elements in at least one of the two sets:
AB = { x | x A x B }
U
AB
A B
42
38. Intersection
Elements in exactly one of the two sets:
AB = { x | x A x B }
U
A A B B
43
39. Disjoint Sets
DEF: If A and B have no common elements, they
are said to be disjoint, i.e. A B = .
U
A B
44
40. Disjoint Union
When A and B are disjoint, the disjoint union
operation is well defined. The circle above the
union symbol indicates disjointedness.
U
A B
A B
45
41. Disjoint Union
FACT: In a disjoint union of finite sets,
cardinality of the union is the sum of the
cardinalities. I.e.
A B A B
46
42. Set Difference
Elements in first set but not second:
A- B = { x | x A x B }
U
A- B B
A
47
43. Symmetric Difference
Elements in exactly one of the two sets:
AB = { x | x A x B }
A B U
A B
48
44. Complement
Elements not in the set (unary operator):
A = { x | x A }
U
A A
49
45. Set Identities
In fact, the logical identities create the set
identities by applying the definitions of the various
set operations. For example:
LEMMA: (Associativity of Unions)
(AB )C = A(B C )
50
46. Set Identities
In fact, the logical identities create the set
identities by applying the definitions of the various
set operations. For example:
LEMMA: (Associativity of Unions)
(AB )C = A(B C )
Proof : (AB )C = {x | x A B x C } (by def.)
51
47. Set Identities
In fact, the logical identities create the set
identities by applying the definitions of the various
set operations. For example:
LEMMA: (Associativity of Unions)
(AB )C = A(B C )
Proof : (AB )C = {x | x A B x C } (by def.)
= {x | (x A x B ) x C } (by def.)
52
48. Set Identities
In fact, the logical identities create the set
identities by applying the definitions of the various
set operations. For example:
LEMMA: (Associativity of Unions)
(AB )C = A(B C )
Proof : (AB )C = {x | x A B x C } (by def.)
= {x | (x A x B ) x C } (by def.)
= {x | x A ( x B x C ) } (logical assoc.)
53
49. Set Identities
In fact, the logical identities create the set
identities by applying the definitions of the various
set operations. For example:
LEMMA: (Associativity of Unions)
(AB )C = A(B C )
Proof : (AB )C = {x | x A B x C } (by def.)
= {x | (x A x B ) x C } (by def.)
= {x | x A ( x B x C ) } (logical assoc.)
= {x | x A x B C ) } (by def.)
54
50. Set Identities
In fact, the logical identities create the set
identities by applying the definitions of the various
set operations. For example:
LEMMA: (Associativity of Unions)
(AB )C = A(B C )
Proof : (AB )C = {x | x A B x C } (by def.)
= {x | (x A x B ) x C } (by def.)
= {x | x A ( x B x C ) } (logical assoc.)
= {x | x A (x B C ) } (by def.)
= A(B C ) (by def.)
Other identities are derived similarly.
55
51. Set Identities via Venn
It’s often simpler to understand an
identity by drawing a Venn Diagram.
For example DeMorgan’s first law
A B A B
can be visualized as follows.
56
59. Sets as Bit-Strings
If we order the elements of our universe, we
can represent sets by bit-strings. For example,
consider the universe
U = {ant, beetle, cicada, dragonfly}
Order the elements alphabetically. Subsets of U
are represented by bit-strings of length 4. Each
bit in turn, tells us whether the corresponding
element is contained in the set. EG: {ant,
dragonfly} is represented by the bit-string
1001.
Q: What set is represented by 0111 ?
64
60. Sets as Bit-Strings
A: 0111 represents
{beetle, cicada, dragonfly}
Conveniently, under this representation the
various set theoretic operations become the
logical bit-string operators that we saw before.
For example, the symmetric difference of
{beetle} with {ant, beetle, dragonfly} is
represented by:
0100
1101
1001 = {ant, dragonfly}
65
61. Relations
In mathematics, a binary relation on a set A is a collection
of ordered pairs of elements of A. It is a subset of the
Cartesian product A2 = A × A.
More generally, a binary relation between two sets A and B
is a subset of A × B.
The terms dyadic relation and 2-place relation are
synonyms for binary relations.
A binary relation is the special case n = 2 of an n-ary
relation R ⊆ A1 × … × An, that is, a set of n-tuples where the
jth component of each n-tuple is taken from the jth domain Aj
of the relation.
66
62. Relations
A binary relation R is usually defined as an ordered triple
<X, Y, G)>
where X and Y are arbitrary sets (or classes), and G is a subset of the
Cartesian product X × Y. The sets X and Y are called the domain (or the
set of departure) and codomain (or the set of destination), respectively, of
the relation, and G is called its graph.
The statement (x,y) ∈ R is read "x is R-related to y", and is denoted by
xRy or R(x,y). The latter notation corresponds to viewing R as the
characteristic function on "X" x "Y" for the set of pairs of G.
The order of the elements in each pair of G is important: if a ≠ b, then
aRb and bRa can be true or false, independently of each other.
67
63. Relations
According to the definition above, two relations with the same
graph may be different, if they differ in the sets X and Y. For
example, if G = {(1,2),(1,3),(2,7)}, then (Z,Z, G), (R, N, G), and
(N, R, G) are three distinct relations.
Suppose there are four objects {ball, car, doll, gun} and four
persons {John, Mary, Ian, Venus}. Suppose that John owns the
ball, Mary owns the doll, and Venus owns the car. The binary
relation "is owned by" is given as
R=<{ball, car, doll, gun}, {John, Mary, Ian, Venus}, {(ball,
John), (doll, Mary), (car, Venus)}>
68
64. Relations
Uniqueness:
injective (left-unique): for all x and z in X and y in Y it holds that if xRy and
zRy then x = z.
functional (right-unique, right-definite): for all x in X, and y and z in Y it
holds that if xRy and xRz then y = z; such a binary relation is called a partial
function.
one-to-one (1-to-1): injective and functional.
Totality:
left-total: for all x in X there exists a y in Y such that xRy.
surjective (right-total): for all y in Y there exists an x in X such that xRy.
A correspondence: a binary relation that is both left-total and surjective.
Uniqueness and totality properties:
A function: a relation that is functional and left-total.
A bijection: a one-to-one correspondence; such a relation is a function and is
said to be bijective.
69
65. Relations
reflexive: for all x in X it holds that xRx. For example, "greater than or equal to" is a
reflexive relation but "greater than" is not.
irreflexive (or strict): for all x in X it holds that not xRx. "Greater than" is an example of
an irreflexive relation.
coreflexive: for all x and y in X it holds that if xRy then x = y. "Equal to" is an example of
a coreflexive relation.
symmetric: for all x and y in X it holds that if xRy then yRx. "Is a blood relative of”
antisymmetric: for all distinct x and y in X, if xRy then not yRx.
asymmetric: for all x and y in X, if xRy then not yRx. (So asymmetricity is stronger than
anti-symmetry. In fact, asymmetry is equivalent to anti-symmetry plus irreflexivity.)
transitive: for all x, y and z in X it holds that if xRy and yRz then xRz. (Note that, under the
assumption of transitivity, irreflexivity and asymmetry are equivalent.)
total: for all x and y in X it holds that xRy or yRx (or both). "Is greater than or equal to" is
an example of a total relation.
70
66. Relations
Binary relations by property
reflexive symmetric transitive symbol example
undirected graph No Yes
dependency Yes Yes
weak order Yes ≤
preorder Yes Yes ≤ preference
partial order Yes No Yes ≤ subset
partial equivalence Yes Yes
equivalence relation Yes Yes Yes ∼, ≅, ≈, ≡ equality
proper
strict partial order No No Yes <
subset
71
68. Agenda
Functions
Domain, co-domain, range
Image, pre-image
One-to-one, onto, bijective, inverse
Functional composition and exponentiation
Ceiling “ ” and floor “ ”
Sequences and Sums
Sequences ai
Summations ai
i 0
Countable 0 and uncountable sets
73
69. Functions
In high-school, functions are often identified
with the formulas that define them.
EG: f (x ) = x 2
This point of view does not suffice in Discrete
Math. In discrete math, functions are not
necessarily defined over the real numbers.
EG: f (x ) = 1 if x is odd, and 0 if x is even.
So in addition to specifying the formula one
needs to define the set of elements which are
acceptable as inputs, and the set of elements
into which the function outputs.
74
70. Functions. Basic-Terms.
DEF: A function f : A B is given by a
domain set A, a codomain set B, and a
rule which for every element a of A,
specifies a unique element f (a) in B. f
(a) is called the image of a, while a is
called the pre-image of f (a). The
range (or image) of f is defined by
f (A) = {f (a) | a A }.
75
71. Functions. Basic-Terms.
EG: Let f : Z R be given by f (x ) = x 2
Q1: What are the domain and co-domain?
Q2: What’s the image of -3 ?
Q3: What are the pre-images of 3, 4?
Q4: What is the range f (Z) ?
76
72. Functions. Basic-Terms.
f : Z R is given by f (x ) = x 2
A1: domain is Z, co-domain is R
A2: image of -3 = f (-3) = 9
A3: pre-images of 3: none as 3 isn’t an
integer!
pre-images of 4: -2 and 2
A4: range is the set of perfect squares
f (Z) = {0,1,4,9,16,25,…}
77
73. One-to-One, Onto, Bijection.
Intuitively.
Represent functions using “node and arrow” notation:
One-to-One means that no clashes occur.
BAD: a clash occurred, not 1-to-1
GOOD: no clashes, is 1-to-1
Onto means that every possible output is hit
BAD: 3rd output missed, not onto
GOOD: everything hit, onto
83
74. One-to-One, Onto, Bijection.
Intuitively.
Bijection means that when arrows reversed,
a function results. Equivalently, that both one-
to-one’ness and onto’ness occur.
BAD: not 1-to-1. Reverse
over-determined:
BAD: not onto. Reverse
under-determined:
GOOD: Bijection. Reverse
is a function:
84
75. One-to-One, Onto, Bijection.
Formal Definition.
DEF: A function f : A B is:
one-to-one (or injective) if different elements of A
always result in different images in B.
onto (or surjective) if every element in B is hit by f.
I.e., f (A ) = B.
a one-to-one correspondence (or a bijection, or
invertible) if f is both one-to-one as well as onto.
If f is invertible, its inverse f -1 : B A is well defined
by taking the unique element in the pre-image of b, for
each b B.
85
76. One-to-One, Onto, Bijection.
Examples.
Q: Which of the following are 1-to-1, onto, a
bijection? If f is invertible, what is its
inverse?
1. f : Z R is given by f (x ) = x 2
2. f : Z R is given by f (x ) = 2x
3. f : R R is given by f (x ) = x 3
4. f : Z N is given by f (x ) = |x |
5. f : {people} {people} is given by
f (x ) = the father of x.
86
77. One-to-One, Onto, Bijection.
Examples.
1. f : Z R, f (x ) = x 2: none
2. f : Z Z, f (x ) = 2x : 1-1
3. f : R R, f (x ) = x 3: 1-1, onto,
bijection, inverse is f (x ) = x (1/3)
4. f : Z N, f (x ) = |x |: onto
5. f (x ) = the father of x : none
87
78. Composition
When a function f spits out elements of the
same kind that another function g eats, f and g
may be composed by letting g immediately eat
each output of f.
DEF: Suppose that g : A B and f : B C
are functions. Then the composite
f g : A C is defined by setting
f g (a) = f ( g (a) )
88
79. Composition. Examples.
Q: Compute g f where
1. f : Z R, f (x ) = x 2
and g : R R, g (x ) = x 3
2. f : Z Z, f (x ) = x + 1
and g = f -1 so g (x ) = x – 1
3. f : {people} {people},
f (x ) = the father of x, and g = f
89
80. Composition. Examples.
1. f : Z R, f (x ) = x 2
and g : R R, g (x ) = x 3
f g : Z R , f g (x ) = x 6
2. f : Z Z, f (x ) = x + 1
and g = f -1
f g (x ) = x (true for any function
composed with its inverse)
3. f : {people} {people},
f (x ) = g(x ) = the father of x
f g (x ) = grandfather of x from father’s side
90
81. Repeated Composition
When the domain and codomain are equal, a
function may be self composed. The
composition may be repeated as much as
desired resulting in functional
exponentiation. The whole process is
denoted by
n
f n (x ) = f f f f … f (x )
where f appears n –times on the right side.
Q1: Given f : Z Z, f (x ) = x 2 find f 4
Q2: Given g : Z Z, g (x ) = x + 1 find g n
Q3: Given h(x ) = the father of x, find hn
91
82. Repeated Composition
A1: f : Z Z, f (x ) = x 2.
f 4(x ) = x (2*2*2*2) = x 16
A2: g : Z Z, g (x ) = x + 1
gn (x ) = x + n
A3: h (x ) = the father of x,
hn (x ) = x ’s n’th patrilineal ancestor
92
83. Ceiling and Floor
This being a course on discrete math, it is often
useful to discretize numbers, sets and functions.
For this purpose the ceiling and floor functions
come in handy.
DEF: Given a real number x : The floor of x is
the biggest integer which is smaller or equal to x
The ceiling of x is the smallest integer greater
or equal to x.
NOTATION: floor(x) = x , ceiling(x) = x
Q: Compute 1.7, -1.7, 1.7, -1.7.
93
84. Ceiling and Floor
A: 1.7 = 1, -1.7 = -2,
1.7 = 2, -1.7 = -1
Q: What’s the difference between the
floor function and the (int) casting
function in Java?
94
85. Ceiling and Floor
A: Casting to int in Java always
truncates towards 0. Ceiling and floor are
not symmetric in this way.
EG: (int)(-1.7) == -1
-1.7 = -2
95
86. Example for section 1.6
Consider the function f : R2 R2
defined by the formula
f (x,y ) = ( ax+by, cx+dy )
where a,b,c,d are constants. Give a
condition on the constants which
guarantees that f is one-to-one.
More detailed example
96
87. Sequences
Sequences are a way of ordering lists of
objects. Java arrays are a type of sequence of
finite size. Usually, mathematical sequences
are infinite.
To give an ordering to arbitrary elements, one
has to start with a basic model of order. The
basic model to start with is the set
N = {0, 1, 2, 3, …} of natural numbers.
For finite sets, the basic model of size n is:
n = {1, 2, 3, 4, …, n-1, n }
97
88. Sequences
DEF: Given a set S, an (infinite) sequence in S is a
function N S. A finite sequence in S is a function
n S.
Symbolically, a sequence is represented using the
subscript notation ai . This gives a way of specifying
formulaically
Note: Other sets can be taken as ordering models.
The book often uses the positive numbers Z+ so
counting starts at 1 instead of 0. I’ll usually assume the
ordering model N.
Q: Give the first 5 terms of the sequence defined by
the formula
π
ai cos( i )
98
2
89. Sequence Examples
A: Plug in for i in sequence 0, 1, 2, 3, 4:
a0 1, a1 0, a2 -1, a3 0, a4 1
Formulas for sequences often represent
patterns in the sequence.
Q: Provide a simple formula for each
sequence:
a) 3,6,11,18,27,38,51, …
b) 0,2,8,26,80,242,728,…
c) 1,1,2,3,5,8,13,21,34,…
99
90. Sequence Examples
A: Try to find the patterns between numbers.
a) 3,6,11,18,27,38,51, …
a1=6=3+3, a2=11=6+5, a3=18=11+7, … and in
general ai +1 = ai +(2i +3). This is actually a good
enough formula. Later we’ll learn techniques that
show how to get the more explicit formula:
ai = 6 + 4(i –1) + (i –1)2
b) 0,2,8,26,80,242,728,…
If you add 1 you’ll see the pattern more clearly.
ai = 3i –1
c) 1,1,2,3,5,8,13,21,34,…
This is the famous Fibonacci sequence given by
ai +1 = ai + ai-1
100
91. Bit Strings
Bit strings are finite sequences of 0’s and 1’s.
Often there is enough pattern in the bit-string
to describe its bits by a formula.
EG: The bit-string 1111111 is described by the
formula ai =1, where we think of the string of
being represented by the finite sequence
a1a2a3a4a5a6a7
Q: What sequence is defined by
a1 =1, a2 =1 ai+2 = ai ai+1
101
92. Bit Strings
A: a0 =1, a1 =1 ai+2 = ai ai+1:
1,1,0,1,1,0,1,1,0,1,…
102
93. Summations
The symbol “S” takes a sequence of numbers
and turns it into a sum.
Symbolically: n
a
i 0
i a0 a1 a2 ... an
This is read as “the sum from i =0 to i =n of ai”
Note how “S” converts commas into plus signs.
One can also take sums over a set of numbers:
x
xS
2
103
94. Summations
EG: Consider the identity sequence
ai = i
Or listing elements: 0, 1, 2, 3, 4, 5,…
The sum of the first n numbers is given
by: n
ai 1 2 3 ... n
i 1
(The first term 0 is dropped)
104
95. Summation Formulas –
Arithmetic
There is an explicit formula for the previous:
n
n(n 1)
i 2
i 1
Intuitive reason: The smallest term is 1, the
biggest term is n so the avg. term is (n+1)/2.
There are n terms. To obtain the formula
simply multiply the average by the number of
terms.
105
96. Summation Formulas –
Geometric
Geometric sequences are number
sequences with a fixed constant of
proportionality r between consecutive
terms. For example:
2, 6, 18, 54, 162, …
Q: What is r in this case?
106
97. Summation Formulas
2, 6, 18, 54, 162, …
A: r = 3.
In general, the terms of a geometric sequence
have the form
ai = a r i
where a is the 1st term when i starts at 0.
A geometric sum is a sum of a portion of a
geometric sequence and has the following
explicit formula:
n 1
n
ar - a
ar a ar ar ... ar r - 1
i 0
i 2 n
107
98. Summation Examples
If you are curious about how one could prove
such formulas, your curiosity will soon be
“satisfied” as you will become adept at
proving such formulas a few lectures from
now!
Q: Use the previous formulas to evaluate each
of the following
1.
103
5(i - 3)
i 20
13
2. 2i
i 0
108
99. Summation Examples
A:
1. Use the arithmetic sum formula and
additivity of summation:
103 103 103 103
5(i - 3) 5 (i - 3) 5 i - 5 3
i 20 i 20 i 20 i 20
(103 20)
5 84 - 5 3 84 24570
2
109
100. Summation Examples
A:
2. Apply the geometric sum formula
directly by setting a = 1 and r = 2:
13
214 - 1 14
i 0
2i
2 -1
2 - 1 16383
110
101. Cardinality and Countability
Up to now cardinality has been the number of
elements in a finite sets. Really, cardinality is a
much deeper concept. Cardinality allows us to
generalize the notion of number to infinite
collections and it turns out that many type of
infinities exist.
EG:
{,}
{ , }
{Ø , {Ø,{Ø,{Ø}}} }
These all share “2-ness”.
111
102. Cardinality and Countability
For finite sets, can just count the elements to
get cardinality. Infinite sets are harder.
First Idea: Can tell which set is bigger by
seeing if one contains the other.
{1, 2, 4} N
{0, 2, 4, 6, 8, 10, 12, …} N
So set of even numbers ought to be smaller
than the set of natural number because of strict
containment.
Q: Any problems with this?
112
103. Cardinality and Countability
A: Set of even numbers is obtained from N by
multiplication by 2. I.e.
{even numbers} = 2•N
For finite sets, since multiplication by 2 is a
one-to-one function, the size doesn’t change.
EG: {1,7,11} – 2 {2,14,22}
Another problem: set of even numbers is
disjoint from set of odd numbers. Which one is
bigger?
113
104. Cardinality and Countability –
Finite Sets
DEF: Two sets A and B have the same
cardinality if there’s a bijection
f:AB
For finite sets this is the same as the old
definition:
{,}
{ , }
114
105. Cardinality and Countability –
Infinite Sets
But for infinite sets…
…there are surprises.
DEF: If S is finite or has the same cardinality as N, S is
called countable.
Notation, the Hebrew letter Aleph is often used to
denote infinite cardinalities. Countable sets are said to
have cardinality . 0
Intuitively, countable sets can be counted in the sense
that if you allocate 1 second to count each member,
eventually any particular member will be counted after
a finite time period. Paradoxically, you won’t be able to
count the whole set in a finite time period!
115
106. Countability – Examples
Q: Why are the following sets countable?
1. {0,2,4,6,8,…}
2. {1,3,5,7,9,…}
100
100100
100
3. {1,3,5,7, 100 }
4. Z
116
107. Countability – Examples
1. {0,2,4,6,8,…}: Just set up the
bijection f (n ) = 2n
2. {1,3,5,7,9,…} : Because of the
bijection f (n ) = 2n100 1
+
100100
3. {1,3,5,7, 100100 } has cardinality
5 so is therefore countable
4. Z: This one is more interesting.
Continue on next page:
117
108. Countability of the Integers
Let’s try to set up a bijection between N and Z.
One way is to just write a sequence down
whose pattern shows that every element is
hit (onto) and none is hit twice (one-to-
one). The most common way is to
alternate back and forth between the
positives and negatives. I.e.:
0,1,-1,2,-2,3,-3,…
It’s possible to write an explicit formula down
for this sequence which makes it easier to
check for bijectivity: i i 1
ai -(-1)
2
118
109. Demonstrating Countability.
Useful Facts
Because 0 is the smallest kind of infinity, it
turns out that to show that a set is countable
one can either demonstrate an injection into N
or a surjection from N.
THM: Suppose A is a set. If there is an one-to-
one function f : A N, or there is an onto
function g : N A then A is countable.
The proof requires the principle of mathematical
induction, which we’ll get to at a later date.
119
110. Uncountable Sets
But R is uncountable (“not countable”)
Q: Why not ?
120
111. Uncountability of R
A: This is not a trivial matter. Here are some
typical reasonings:
1. R strictly contains N so has bigger
cardinality. What’s wrong with this
argument?
2. R contains infinitely many numbers
between any two numbers. Surprisingly,
this is not a valid argument. Q has the
same property, yet is countable.
3. Many numbers in R are infinitely complex in
that they have infinite decimal expansions.
An infinite set with infinitely complex
numbers should be bigger than N.
121
112. Uncountability of R
Last argument is the closest.
Here’s the real reason: Suppose that R were
countable. In particular, any subset of R,
being smaller, would be countable also. So
the interval [0,1] would be countable. Thus
it would be possible to find a bijection from
Z+ to [0,1] and hence list all the elements
of [0,1] in a sequence.
What would this list look like?
r1 , r2 , r3 , r4 , r5 , r6 , r7, …
122
113. Uncountability of R
Cantor’s Diabolical Diagonal
So we have this list
r1 , r2 , r3 , r4 , r5 , r6 , r7, …
supposedly containing every real number
between 0 and 1.
Cantor’s diabolical diagonalization
argument will take this supposed list,
and create a number between 0 and 1
which is not on the list. This will
contradict the countability assumption
hence proving that R is not countable.
123
123. Uncountability of R
Cantor’s Diabolical Diagonal
GENERALIZE: To construct a number not on
the list “revil”, let ri,j be the j ’th decimal digit
in the fractional part of ri.
Define the digits of revil by the following rule:
The j ’th digit of revil is 5 if ri,j 5. Otherwise
the j’ ’th digit is set to be 4.
This guarantees that revil is an anti-diagonal.
I.e., it does not share any elements on the
diagonal. But every number on the list
contains a diagonal element. This proves
that it cannot be on the list and contradicts
our assumption that R was countable so the
list must contain revil. //QED
133
124. Impossible Computations
Notice that the set of all bit strings is countable. Here’s
how the list looks:
0,1,00,01,10,11,000,001,010,011,100,101,110,111,0000,…
DEF: A decimal number
0.d1d2d3d4d5d6d7…
Is said to be computable if there is a computer program
that outputs a particular digit upon request.
EG:
1. 0.11111111…
2. 0.12345678901234567890…
3. 0.10110111011110….
134
125. Impossible Computations
CLAIM: There are numbers which cannot be computed
by any computer.
Proof : It is well known that every computer program
may be represented by a bit-string (after all, this is
how it’s stored inside). Thus a computer program
can be thought of as a bit-string. As there are
0 bit-strings yet R is uncountable, there can be
no onto function from computer programs to
decimal numbers. In particular, most numbers do
not correspond to any computer program so are
incomputable!
135
126. Blackboard Exercises
Evaluate the double summation:
2 3
ij
i 0 j 1
Show that if A is uncountable and B is
countable then A-B is uncountable.
136
