Chapter 2:
Basic Structures: Sets, Functions,
Sequences and Sums

Sets (Section 2.1)

Set Operations (Section 2.2)

Functions (Section 2.3)

Sequences and Summations
(Section 2.4)
© by Kenneth H. Rosen, Discrete Mathematics & its Applications, Sifth Edition, Mc Graw-Hill, 2007
1
Sets (2.1)

A set is a collection or group of objects or elements
or members. (Cantor 1895)


A set is said to contain its elements.
There must be an underlying universal set U, either
specifically stated or understood.
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions
2
Sets (2.1) (cont.)

Notation:



list the elements between braces:
S = {a, b, c, d}={b, c, a, d, d}
(Note: listing an object more than once does not change the set.
Ordering means nothing.)
specification by predicates:
S= {x| P(x)},
S contains all the elements from U which make the predicate P
true.
brace notation with ellipses:
S = { . . . , -3, -2, -1},
the negative integers.
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions
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Sets (2.1) (cont.)

Common Universal Sets





R = reals
N = natural numbers = {0,1, 2, 3, . . . }, the counting
numbers
Z = all integers = {. . , -3, -2, -1, 0, 1, 2, 3, 4, . .}
Z+ is the set of positive integers
Notation:
x is a member of S or x is an element of S:
x  S.
x is not an element of S:
x  S.
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions
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Sets (2.1) (cont.)

Subsets


Definition: The set A is a subset of the set B, denoted
A  B, iff
x [x  A  x  B]
Definition: The void set, the null set, the empty set,
denoted , is the set with no members.
Note: the assertion x   is always false. Hence
x [x    x  B]
is always true(vacuously). Therefore,  is a subset of
every set.
Note: A set B is always a subset of itself.
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions
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Sets (2.1) (cont.)



Definition: If A  B but A  B the we say A is a proper
subset of B, denoted A  B (in some texts).
Definition: The set of all subset of a set A, denoted P(A),
is called the power set of A.
Example: If A = {a, b} then
P(A) = {, {a}, {b}, {a,b}}
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions
6
Sets (2.1) (cont.)

Definition: The number of (distinct) elements in A,
denoted |A|, is called the cardinality of A.
If the cardinality is a natural number (in N), then the
set is called finite, else infinite.

Example: A = {a, b},
|{a, b}| = 2,
|P({a, b})| = 4.
A is finite and so is P(A).
Useful Fact: |A|=n implies |P(A)| = 2n
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions
7
Sets (2.1) (cont.)

N is infinite since |N| is not a natural number. It is called a transfinite
cardinal number.

Note: Sets can be both members and subsets of other sets.

Example:
A = {,{}}.
A has two elements and hence four subsets:
, {}, {{}}. {,{}}
Note that  is both a member of A and a subset of A!


Russell's paradox: Let S be the set of all sets which are not members
of themselves. Is S a member of itself?
Another paradox: Henry is a barber who shaves all people who do
not shave themselves. Does Henry shave himself?
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions
8
Sets (2.1) (cont.)

Definition: The Cartesian product of A with B, denoted
A x B, is the set of ordered pairs {<a, b> | a  A  b  B}
Notation: n
 Ai   a1 , a 2 ,...,a n  a i  Ai 
i 1
Note: The Cartesian product of anything with  is . (why?)


Example:
A = {a,b}, B = {1, 2, 3}
AxB = {<a, 1>, <a, 2>, <a, 3>, <b, 1>, <b, 2>, <b, 3>}
What is BxA? AxBxA?
If |A| = m and |B| = n, what is |AxB|?
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions
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Set Operations (2.2) (cont.)

Propositional calculus and set theory are both
instances of an algebraic system called a
Boolean Algebra.
The operators in set theory are defined in terms of
the corresponding operator in propositional
calculus
As always there must be a universe U. All sets are
assumed to be subsets of U
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions
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Set Operations (2.2) (cont.)

Definition:
Two sets A and B are equal, denoted A = B, iff
x [x  A  x  B].

Note: By a previous logical equivalence we have
A = B iff x [(x  A  x  B)  (x  B  x  A)]
or
A = B iff A  B and B  A
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions
Set Operations (2.2) (cont.)

11
Definitions:

The union of A and B, denoted A U B, is the set {x | x  A  x  B}

The intersection of A and B, denoted A  B, is the set
{x | x  A  x  B}
Note: If the intersection is void, A and B are said to be disjoint.



The complement of A, denoted A , is the set {x | (x  A)}
Note: Alternative notation is Ac, and {x|x  A}.
The difference of A and B, or the complement of B relative to A, denoted
A - B, is the set A  B
Note: The (absolute) complement of A is U - A.
The symmetric difference of A and B, denoted A  B, is the set
(A - B) U (B - A)
12
Set Operations (2.2) (cont.)

Examples:
U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
A= {1, 2, 3, 4, 5},
B = {4, 5, 6, 7, 8}. Then







AB = {1, 2, 3, 4, 5, 6, 7, 8}
A  B = {4, 5}
A = {0, 6, 7, 8, 9, 10}
B = {0, 1, 2, 3, 9, 10}
A - B = {1, 2, 3}
B - A = {6, 7, 8}
AB = {1, 2, 3, 6, 7, 8}
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions
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Set Operations (2.2) (cont.)

Venn Diagrams




A useful geometric visualization tool (for 3 or less sets)
The Universe U is the rectangular box
Each set is represented by a circle and its interior
All possible combinations of the sets must be represented
U
U
A
B
For 2 sets

B
A
C
For 3 sets
Shade the appropriate region to represent the given set operation.
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions
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Set Operations (2.2) (cont.)

Set Identities


Set identities correspond to the logical equivalences.
Example:
The complement of the union is the intersection of the
complements:
A B = A B
Proof: To show:
x [x  A  B  x  A  B ]
To show two sets are equal we show for all x that x is a
member of one set if and only if it is a member of the
other.
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions
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Set Operations (2.2) (cont.)

We now apply an important rule of inference (defined
later) called
Universal Instantiation
In a proof we can eliminate the universal quantifier
which binds a variable if we do not assume anything
about the variable other than it is an arbitrary member of
the Universe. We can then treat the resulting predicate
as a proposition.
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions

We say
16
'Let x be arbitrary.'
Then we can treat the predicates as propositions:
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions
17
Set Operations (2.2) (cont.)
Hence
x  A B  x  A  B
is a tautology.
Since
• x was arbitrary
• we have used only logically equivalent assertions and
definitions
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions
18
Set Operations (2.2) (cont.)
we can apply another rule of inference called
Universal Generalization
We can apply a universal quantifier to bind a variable if we
have shown the predicate to be true for all values of the
variable in the Universe.
and claim the assertion is true for all x, i.e.,
x [x  A  B  x  A  B ]
Q. E. D. (Latin phrase “Quod Erat Demonstrandum”)
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions
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Set Operations (2.2) (cont.)


Note: As an alternative which might be easier in some cases, use the
identity
A = B  [A  B and B  A]
Example:
Show A  (B - A) = 
The void set is a subset of every set. Hence,
A  (B - A)  
Therefore, it suffices to show
A  (B - A)  
or
x [xA  (B - A)  x  ]
So as before we say 'let x be arbitrary’.
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions
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Set Operations (2.2) (cont.)

Example (cont.)
Show xA  (B - A)  x   is a tautology.
But the consequent is always false.
Therefore, the antecedent better always be false also.
Apply the definitions:
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions
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Set Operations (2.2) (cont.)

Example (cont.)
Hence, because P  P is always false, the implication
is a tautology.
The result follows by Universal Generalization.
Q. E. D.
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions
22
Set Operations (2.2) (cont.)

Union and Intersection of Indexed Collections


Let A1,A2 ,..., An be an indexed collection of sets.
Union and intersection are associative (because 'and' and
'or' are) we have:
n
 Ai  A1  A2  ...  An
i 1
and
n
 Ai  A1  A2  ...  An
i 1
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions
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Set Operations (2.2) (cont.)

Examples
Let
Ai  [ iA, 
),
1

i


 [ i ,  ),1  i  
i
n
n
 Ai  [ 1 ,  )
 Ai i n1 [ 1 ,  )
i 1
n
 Ai  [ n ,  )
i 1
 Ai  [ n ,  )
i 1
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions
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Functions (2.3)

Definition:
Let A and B be sets. A function (mapping, map) f
from A to B, denoted f :AB, is a subset of A*B
such that
x [x  A  y [y  B  < x, y > f ]]
and
[< x, y1 > f  < x, y2 >  f ]  y1 = y2
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions
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Functions (2.3) (cont.)

Note: f associates with each x in A one and only one y in B.
A is called the domain and
B is called the codomain.
If f(x) = y
y is called the image of x under f
x is called a preimage of y
(note there may be more than one preimage of y but there
is only one image of x).
The range of f is the set of all images of points in A under
f. We denote it by f(A).
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions
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Functions (2.3) (cont.)
If S is a subset of A then
f(S) = {f(s) | s in S}.
Example:
A
•
•
•
•
•
•
•
•
f(a) = Z
the image of d is Z
the domain of f is A = {a, b, c, d}
the codomain is B = {X, Y, Z}
f(A) = {Y, Z}
the preimage of Y is b
the preimages of Z are a, c and d
f({c,d}) = {Z}
B
a
b
c
d
X
Y
Z
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions
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Functions (2.3) (cont.)

Injections, Surjections and Bijections




Let f be a function from A to B.
Definition: f is one-to-one (denoted 1-1) or injective if preimages
are unique.
Note: this means that if a  b then f(a)  f(b).
Definition: f is onto or surjective if every y in B has a preimage.
Note: this means that for every y in B there must be an x in A such
that f(x) = y.
Definition: f is bijective if it is surjective and injective (one-to-one
and onto).
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions
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Functions (2.3) (cont.)

Examples:
The previous Example function is neither an injection nor
a surjection. Hence it is not a bijection.
A
B
a
b
c
d
X
Y
Z
Surjection but not an injection
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions
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Functions (2.3) (cont.)
A
a
b
c
d
A
B
a
V
b
c
W
X
d
B
V
W
X
Y
Injection & a surjection,
hence a bijection
Y
Z
Injection but not a surjection
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions
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Functions (2.3) (cont.)


Note: Whenever there is a bijection from A to B, the two
sets must have the same number of elements
or the same cardinality.
That will become our definition, especially for infinite
sets.
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions
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Functions (2.3) (cont.)

Examples:
Let A = B = R, the reals. Determine which are
injections, surjections, bijections:
•
•
•
•
•
f(x)
f(x)
f(x)
f(x)
f(x)
=
=
=
=
=
x,
x2,
x3,
x + sin(x),
|x|
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions
32
Functions (2.3) (cont.)

Let E be the set of even integers {0, 2, 4, 6, . . . .}.
Then there is a bijection f from N to E , the even
nonnegative integers, defined by
f(x) = 2x.
Hence, the set of even integers has the same cardinality
as the set of natural numbers.
OH, NO! IT CAN’T BE....E IS ONLY HALF AS BIG!!!
Sorry! It gets worse before it gets better.
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions
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Functions (2.3) (cont.)

Inverse Functions

Definition:
Let f be a bijection from A to B. Then the inverse of f,
denoted f-1, is the function from B to A defined as
f-1(y) = x iff f(x) = y
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions
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Functions (2.3) (cont.)

Example: Let f be defined by the diagram:
A
a
b
c
d
f
A
B
f-1
a
V
W
X
b
c
d
Y
B
V
W
X
Y
Note: No inverse exists
unless f is a bijection
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions
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Functions (2.3) (cont.)


Definition: Let S be a subset of B. Then
f-1(S) = {x | f(x)  S}
Note: f need not be a bijection for this definition to hold.
Example: Let f be the following function:
A
B
a
b
c
d
X
Y
Z
f-1({Z}) = {c, d}
f-1({X, Y}) = {a, b}
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions
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Functions (2.3) (cont.)

Composition

Definition:
Let f: B C, g: A B. The composition of f with g,
denoted fg, is the function from A to C defined by
f  g(x) = f(g(x))
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions
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A

Examples:
g
B
f
C
a
V
b
c
W
i
X
j
d
h
Y
A
a
b
c
d
fg
C
h
i
j
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions
38
Functions (2.3) (cont.)


If f(x) = x2 and g(x) = 2x + 1, then f(g(x)) = (2x+1)2
and g(f(x)) = 2x2 + 1
Definition:


The floor function, denoted f ( x) = x or
f(x) = floor(x), is the largest integer less than or equal to x.
The ceiling function, denoted f ( x) = x or f(x) = ceiling(x), is
the smallest integer greater than or equal to x.
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions
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Functions (2.3) (cont.)


Examples: 3.5 = 3, 3.5 = 4.
Note: the floor function is equivalent to truncation for positive numbers.
Example:
Suppose f: B  C, g: A  B and f  g is injective.
What can we say about f and g?
• We know that if a  b then f(g(a))  f(g(b)) since the composition is
injective.
• Since f is a function, it cannot be the case that g(a) = g(b) since then f
would have two different images for the same point.
• Hence, g(a)  g(b)
It follows that g must be an injection.
However, f need not be an injection (you show).
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions
40
Sequences and Summations (2.4)

Definition: A sequence is a function from a subset of the
natural numbers (usually of the form {0, 1, 2, . . . } to a set
S.
Note: the sets
{0, 1, 2, 3, . . . , k} and {1, 2, 3, 4, . . . , k}
are called initial segments of N.
Notation: if f is a function from {0, 1, 2, . . .} to S we
usually denote f(i) by ai and we write
a0 , a1 , a 2 ,...  a i ki 0  a i 0k
where k is the upper limit (usually ).
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions
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Sequences and Summations (2.4) (cont.)
Examples:
Using zero-origin indexing, if f(i) = 1/(i + 1). then the
Sequence
f = {1, 1/'2,1/3,1/4, . . . } = {a0, a1, a2, a3, . . }
Using one-origin indexing the sequence f becomes
{1/2, 1/3, . . .} = {a1, a2, a3, . . .}
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions
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Sequences and Summations (2.4) (cont.)

Summation Notation
 
k
Given a sequence a i 0 we can add together a subset
of the sequence by using the summation and function
notation
a g ( m )  a g ( m  1 )  ...  a g ( n ) 
or more generally
n
 a g( j )
j m
aj
jS
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions
43
Sequences and Summations (2.4) (cont.)
n
Examples: r 0  r 1  r 2  ...  r n   r j
0

1 1 1
1
1     ...  
2 3 4
1 i
a 2 m  a 2 ( m  1 )  ...  a 2 ( n ) 
n
 a2 j
jm
if S  {2,5,7,10} then  a j  a 2  a 5  a7  a 10
jS
n
Similarity for the product notation:
 a j  a m a m  1 ...a n
j m
44
Sequences and Summations (2.4) (cont.)
Definition:
A geometric progression is a sequence of the form
a, ar, ar2, ar3, ar4, . . . .
Your book has a proof that
n 1
n
r
i

r
i 0
1
if r  1
r 1
(you can figure out what it is if r = 1).
You should also be able to determine the sum
• if the index starts at k vs. 0
• if the index ends at something other than n
(e.g., n-1, n+1, etc.).
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions
45
Sequences and Summations (2.4) (cont.)

Cardinality

Definition:
The cardinality of a set A is equal to the cardinality of a
set B, denoted
| A | = | B |,
if there exists a bijection from A to B.
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions
46
Sequences and Summations (2.4)

Definition:
If a set has the same cardinality as a subset of the
natural numbers N, then the set is called countable.
If |A| = |N|, the set A is countably infinite.
The (transfinite) cardinal number of the set N is
aleph null = 0.
If a set is not countable we say it is uncountable.
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions
47
Sequences and Summations (2.4)

Examples:
The following sets are uncountable (we show later)



The real numbers in [0, 1]
P(N), the power set of N
Note: With infinite sets proper subsets can have the
same cardinality. This cannot happen with finite sets.
Countability carries with it the implication that there is a
listing of the elements of the set.
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions
48
Sequences and Summations (2.4)

Definition: | A |  | B | if there is an injection from A to B.
Note: as you would hope,

Theorem:
If | A |  | B | and | B |  | A | then | A | = | B |.
This implies
 if there is an injection from A to B
 if there is an injection from B to A
then
 there must be a bijection from A to B


This is difficult to prove but is an example of demonstrating existence
without construction.
It is often easier to build the injections and then conclude the bijection
exists.
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions
49
Sequences and Summations (2.4)

Example:
Theorem: If A is a subset of B then | A |  | B |.
Proof: the function f(x) = x is an injection from A to B.

Example: {0, 2, 5}|  0
The injection f: {0, 2, 5}  N defined by f(x) = x is
shown below:
0 1 2 3 4 5 6…
0
2
5
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Sequences and Summations (2.4)

Some Countably Infinite Sets

The set of even integers E ( 0 is considered even) is
countably infinite. Note that E is a proper subset of N,
Proof: Let f(x) = 2x. Then f is a bijection from N to E
0 1 2 3 4 5
6…

0 2 4 6 8 10 12 …
Z+, the set of positive integers is countably infinite.
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Sequences and Summations (2.4)

The set of positive rational numbers Q+ is countably
infinite.
Proof: Z+ is a subset of Q+ so |Z+| = 0  |Q+|.
Now we have to show that |Q+|  0.
To do this we show that the positive rational numbers
with repetitions, QR, is countably infinite.
Then, since Q+ is a subset of QR, it follows that
|Q+|  0 and hence |Q+| = 0.
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
The position on the path (listing) indicates the image of
the bijective function f from N to QR:
f(0) = 1/1, f(1) = 1/2, f(2) = 2/1, f(3) = 3/1, and so
forth.
Every rational number appears on the list at least once,
some many times (repetitions).
Hence, |N| = |QR| = 0.

Q. E. D
The set of all rational numbers Q, positive and negative, is
countably infinite.
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions
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Sequences and Summations (2.4)

The set of (finite length) strings S over a finite alphabet A is
countably infinite.
To show this we assume that


A is nonvoid
There is an “alphabetical” ordering of the symbols in A
Proof: List the strings in lexicographic order:
all the strings of zero length,
 then all the strings of length 1 in alphabetical order,
 then all the strings of length 2 in alphabetical order,
etc.

This implies a bijection from N to the list of strings and hence it is a
countably infinite set.
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions
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Sequences and Summations (2.4)

For example:
Let A = {a, b, c}.
Then the lexicographic ordering of A is
{ , a, b, c, aa, ab, ac, ba, bb, bc, ca, cb, cc, aaa, aab,
aac, aba, ....} = {f(0), f(1), f(2), f(3), f(4), . . . .}
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Sequences and Summations (2.4)

The set of all C programs is countable.
Proof: Let S be the set of legitimate characters which can appear in
a C program.


A C compiler will determine if an input program is a syntactically correct
C program (the program doesn't have to do anything useful).
Use the lexicographic ordering of S and feed the strings into the
compiler.
– If the compiler says YES, this is a syntactically correct C program, we add
the program to the list.
– Else we move on to the next string.


In this way we construct a list or an implied bijection from N to the
set of C programs.
Hence, the set of C programs is countable.
Q. E. D.
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions
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Sequences and Summations (2.4)

Cantor Diagonalization

An important technique used to construct an object which is not a
member of a countable set of objects with (possibly) infinite
descriptions
Theorem: The set of real numbers between 0 and 1 is uncountable.
Proof: We assume that it is countable and derive a contradiction.
If it is countable we can list them (i.e., there is a bijection from a
subset of N to the set).
We show that no matter what list you produce we can construct a
real number between 0 and 1 which is not in the list.
Hence, there cannot exist a list and therefore the set is not
countable
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It's actually much bigger than countable. It is said to have
the cardinality of the continuum, c.
Represent each real number in the list using its decimal
expansion.
e.g., 1/3 = .3333333........
1/2 = .5000000........
= .4999999........
If there is more than one expansion for a number, it
doesn't matter as long as our construction takes this into
account.
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Sequences and Summations (2.4)

THE LIST....
r1 = .d11d12d13d14d15d16. . . . .
r2 = .d21d22d23d24d25d26 . . . .
r3 = .d31d32d33d34d35d36 . . . .
...
Now construct the number x = .x1x2x3x4x5x6x7. . . .
xi = 3 if dii  3
xi = 4 if dii = 3
(Note: choosing 0 and 9 is not a good idea because of the non uniqueness
of decimal expansions.)
Then x is not equal to any number in the list.
Hence, no such list can exist and hence the interval (0,1) is uncountable.
Q. E. D.
60
Sequences and Summations (2.4)

An extra goody:
Definition: a number x between 0 and 1 is computable
if there is a C program which when given the input i, will
produce the ith digit in the decimal expansion of x.

Example:
The number 1/3 is computable.
The C program which always outputs the digit 3,
regardless if the input, computes the number.
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Theorem: There is exists a number x between 0 and 1
which is not computable.
There does not exist a C program (or a program in any
other language) which will compute it!
Why? Because there are more numbers between 0 and 1
than there are C programs to compute them.
(in fact there are c such numbers!)
Our second example of the nonexistence of programs to
compute things!
CSE 504, Ch.1 (part 3): The foundations: Logic & Proof, Sets, and Functions