Discrete Structures

Dr. S.S.Shehaby

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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!

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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 }

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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 }

{ , , } = { , , }

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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.

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Standard Numerical Sets

A: number irrationalany or , π,,2 e

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-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. 3 R

2. -3 N

3. -3 R

4. 0 Z+

5. x xR x2=-5

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-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.

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-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

S T

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:

S T x (xS ) (xT )

We already had all the necessary concepts, but the ―‖ notation saves work.

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-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:

S T S T x (x S xT )

Q: What algebraic symbol is reminiscent of?

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-Notation

A: is to , as < is to .

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The Empty Set

The empty set is the set containing no elements. This set is also called the null set and is denoted by:

{}

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Subset Examples

Q: Which of the following are true:

1. N R

2. Z N

3. -3 R

4. {1,2} Z+

5.

6.

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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.

6. . No set can contain itself properly.

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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. |{ {}, {{}}, {{{}}} }|

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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

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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.

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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|

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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|

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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.

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Ordered n-tuples

As opposed to sets, repetition and ordering do matter with n-tuples.

(11, 11, 11, 12, 13) ( 11, 12, 13 )

( , , ) ( , , )

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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.

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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 ?

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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?

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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.

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Blackboard Exercise

Prove the following:

If A B and B C then A C .

Set Operations

Agenda Set Operations

Union and Disjoint union

Intersection

Difference “-”

Complement “ ”

Symmetric Difference

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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.

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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.

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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‖.

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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.

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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 | } = ?

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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.

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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 .

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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.

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Union

AB = { x | x A x B }

Elements in at least one of the two sets:

A B

U

AB

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Intersection

AB = { x | x A x B }

Elements in exactly one of the two sets:

A B

U

AB

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Disjoint Sets

DEF: If A and B have no common elements, they

are said to be disjoint, i.e. A B = .

A B

U

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Disjoint Union When A and B are disjoint, the disjoint union operation is well defined. The circle above the union symbol indicates disjointedness.

A B

U

BA

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Disjoint Union

FACT: In a disjoint union of finite sets, cardinality of the union is the sum of the cardinalities. I.e.

BABA

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Set Difference

A-B = { x | x A x B }

Elements in first set but not second:

A

B

U

A-B

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Symmetric Difference

A B

U AB

AB = { x | x A x B }

Elements in exactly one of the two sets:

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Complement

A = { x | x A }

Elements not in the set (unary operator):

A

U

A

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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 )

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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.)

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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.)

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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.)

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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.)

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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.

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Set Identities via Venn

It’s often simpler to understand an

identity by drawing a Venn Diagram.

For example DeMorgan’s first law

can be visualized as follows. BABA

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Visual DeMorgan

A: B:

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Visual DeMorgan

A: B:

AB :

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Visual DeMorgan

A: B:

AB :

:BA

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Visual DeMorgan

A: B:

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Visual DeMorgan

A: B:

A: B:

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Visual DeMorgan

A: B:

A: B:

:BA

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Visual DeMorgan

= BA

BA

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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 ?

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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}

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 × … × A

n, 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.

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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.

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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)}>

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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.

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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.

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Relations

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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

strict partial order No No Yes < proper

subset

Functions; Sequences, Sums, Countability

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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

Countable and uncountable sets

0i

ia

0

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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.

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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 }.

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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) ?

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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,…}

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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

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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:

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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.

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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.

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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

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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) )

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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

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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

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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

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

n

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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

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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.

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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?

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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

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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

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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 }

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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

)2

πcos( iai

99

Sequence Examples

A: Plug in for i in sequence 0, 1, 2, 3, 4:

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,…

1,0,1,0,1 43210 - aaaaa

100

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

101

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

102

Bit Strings

A: a0 =1, a1 =1 ai+2 = ai ai+1:

1,1,0,1,1,0,1,1,0,1,…

103

Summations The symbol “S” takes a sequence of numbers and turns it into a sum.

Symbolically:

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:

n

n

i

i aaaaa

...210

0

Sx

x2

104

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:

(The first term 0 is dropped)

nan

i

i

...3211

105

Summation Formulas –Arithmetic

There is an explicit formula for the previous:

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.

2

)1(

1

nni

n

i

106

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?

107

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:

1...

12

0 -

-

r

aararararaar

nn

n

i

i

108

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.

2.

-103

20

)3(5i

i

13

0

2i

i

109

Summation Examples

A:

1. Use the arithmetic sum formula and additivity of summation:

---103

20

103

20

103

20

103

20

355)3(5)3(5i iii

iii

2457084352

)20103(845 -

110

Summation Examples

A:

2. Apply the geometric sum formula directly by setting a = 1 and r = 2:

163831212

122 14

1413

0

--

-

i

i

111

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”.

112

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?

113

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?

114

Cardinality and Countability – Finite Sets

DEF: Two sets A and B have the same cardinality if there’s a bijection

f : A B

For finite sets this is the same as the old definition:

{,}

{ , }

115

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 .

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!

0

116

Countability – Examples

Q: Why are the following sets countable?

1. {0,2,4,6,8,…}

2. {1,3,5,7,9,…}

3. {1,3,5,7, }

4. Z

100100100100100

117

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 ) = 2n + 1

3. {1,3,5,7, } has cardinality 5 so is therefore countable

4. Z: This one is more interesting. Continue on next page:

100100100100100

118

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:

--

2

1)1(

ia i

i

119

Demonstrating Countability. Useful Facts

Because 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.

0

120

Uncountable Sets

But R is uncountable (“not countable”)

Q: Why not ?

121

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.

122

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, …

123

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.

Cantor's Diagonalization Argument

r1 0.

r2 0.

r3 0.

r4 0.

r5 0.

r6 0.

r7 0.

:

revil 0.

Decimal expansions of ri

Cantor's Diagonalization Argument

r1 0. 1 2 3 4 5 6 7

r2 0.

r3 0.

r4 0.

r5 0.

r6 0.

r7 0.

:

revil 0.

Decimal expansions of ri

Cantor's Diagonalization Argument

r1 0. 1 2 3 4 5 6 7

r2 0. 1 1 1 1 1 1 1

r3 0.

r4 0.

r5 0.

r6 0.

r7 0.

:

revil 0.

Decimal expansions of ri

Cantor's Diagonalization Argument

r1 0. 1 2 3 4 5 6 7

r2 0. 1 1 1 1 1 1 1

r3 0. 2 5 4 2 0 9 0

r4 0.

r5 0.

r6 0.

r7 0.

:

revil 0.

Decimal expansions of ri

Cantor's Diagonalization Argument

r1 0. 1 2 3 4 5 6 7

r2 0. 1 1 1 1 1 1 1

r3 0. 2 5 4 2 0 9 0

r4 0. 7 8 9 0 6 2 3

r5 0.

r6 0.

r7 0.

:

revil 0.

Decimal expansions of ri

Cantor's Diagonalization Argument

r1 0. 1 2 3 4 5 6 7

r2 0. 1 5 1 1 1 1 1

r3 0. 2 5 4 2 0 9 0

r4 0. 7 8 9 0 6 2 3

r5 0. 0 1 1 0 1 0 1

r6 0.

r7 0.

:

revil 0.

Decimal expansions of ri

Cantor's Diagonalization Argument

r1 0. 1 2 3 4 5 6 7

r2 0. 1 5 1 1 1 1 1

r3 0. 2 5 4 2 0 9 0

r4 0. 7 8 9 0 6 2 3

r5 0. 0 1 1 0 1 0 1

r6 0. 5 5 5 5 5 5 5

r7 0.

:

revil 0.

Decimal expansions of ri

Cantor's Diagonalization Argument

r1 0. 1 2 3 4 5 6 7

r2 0. 1 5 1 1 1 1 1

r3 0. 2 5 4 2 0 9 0

r4 0. 7 8 9 0 6 2 3

r5 0. 0 1 1 0 1 0 1

r6 0. 5 5 5 5 5 5 5

r7 0. 7 6 7 9 5 4 4

:

revil 0.

Decimal expansions of ri

Cantor's Diagonalization Argument

r1 0. 1 2 3 4 5 6 7

r2 0. 1 5 1 1 1 1 1

r3 0. 2 5 4 2 0 9 0

r4 0. 7 8 9 0 6 2 3

r5 0. 0 1 1 0 1 0 1

r6 0. 5 5 5 5 5 5 5

r7 0. 7 6 7 9 5 4 4

:

revil 0. 5 4 5 5 5 4 5

Decimal expansions of ri

133

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

134

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….

135

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

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!

0

136

Blackboard Exercises

Evaluate the double summation:

Show that if A is uncountable and B is countable then A-B is uncountable.

2

0

3

1i j

ij