1. © Art Traynor 2011
Mathematics
Definition
Mathematics
Wiki: “ Mathematics ”
1564 – 1642
Galileo Galilei
Grand Duchy of Tuscany
( Duchy of Florence )
City of Pisa
Mathematics – A Language
“ The universe cannot be read until we have learned the language and
become familiar with the characters in which it is written. It is written
in mathematical language…without which means it is humanly
impossible to comprehend a single word.
Without these, one is wandering about in a dark labyrinth. ”

2. © Art Traynor 2011
Mathematics
Definition
Algebra – A Mathematical Grammar
Mathematics
A formalized system ( a language ) for the transmission of
information encoded by number
Algebra
A system of construction by which
mathematical expressions are well-formed
Expression
Symbol Operation Relation
Designate expression
elements or Operands
( Terms / Monomials )
Transformations or LOC’s
capable of rendering an
expression into a relation
A mathematical Structure
between operands
represented by a well-formed
Expression
A well-formed symbolic representation of Operands ( Terms or Monomials ) ,
of discrete arity, upon which one or more Operations ( Laws of Composition - LOC’s )
may structure a Relation
1. Identifies the explanans
by non-tautological
correspondences
Definition
2. Isolates the explanans
as a proper subset from
its constituent
correspondences
3. Terminology
a. Maximal parsimony
b. Maximal syntactic
generality
4. Examples
a. Trivial
b. Superficial
Mathematics
Wiki: “ Polynomial ”
Wiki: “ Degree of a Polynomial ”

3. © Art Traynor 2011
Mathematics
Disciplines
Algebra
One of the disciplines within the field of Mathematics
Mathematics
Others are Arithmetic, Geometry,
Number Theory, & Analysis

The study of expressions of symbols ( sets ) and the
well-formed rules by which they might be consistently
manipulated.

Algebra
Elementary Algebra
Abstract Algebra
A class of Structure defined by the object Set and
its Operations ( or Laws of Composition – LOC’s )

Linear Algebra
Mathematics

4. © Art Traynor 2011
Mathematics
Definitions
Expression
Symbol Operation Relation
Designate expression
elements or Operands
( Terms / Monomials )
Transformations or LOC’s
capable of rendering an
expression into a relation
A mathematical structure
between operands represented
by a well-formed expression
A well-formed symbolic representation of Operands ( Terms or Monomials ) ,
of discrete arity, upon which one or more Operations ( LOC’s ) may structure a Relation
Expression – A Mathematical Sentence
Proposition
A declarative expression
asserting a fact, the truth
value of which can be
ascertained
Formula
A concise symbolic
expression positing a relation
VariablesConstants
An alphabetic character
representing a number the
value of which is arbitrary,
unspecified, or unknown
Operands ( Terms / Monomials )
A transformation
invariant scalar quantity
Mathematics
Predicate
A Proposition admitting the
substitution of variables
O’Leary, Section 2.1,
Pg. 41
Expression constituents consisting of Constants and
Variables exhibiting exclusive parity
Polynomial
An Expression composed of Constants ( Coefficients ) and Variables ( Unknowns) with
an LOC’s of Addition, Subtraction, Multiplication and Non-Negative Exponentiation
Wiki: “ Polynomial ”
Wiki: “ Degree of a Polynomial ”

5. © Art Traynor 2011
Mathematics
Definitions
Expression
Symbol Operation Relation
Designate expression
elements or Operands
( Terms / Monomials )
Transformations capable of
rendering an expression
into a relation
A mathematical structure between operands represented
by a well-formed expression
Expression – A Mathematical Sentence
Proposition
A declarative expression
the truth value of which can
be ascertained
Formula
A concise symbolic
expression positing a relation
VariablesConstants
An alphabetic character
representing a number the
value of which is arbitrary,
unspecified, or unknown
Operands ( Terms / Monomials )
A transformation
invariant scalar quantity
Equation
A formula stating an
equivalency class relation
Inequality
A formula stating a relation
among operand cardinalities
Function
A Relation between a Set of inputs and a Set of permissible
outputs whereby each input is assigned to exactly one output
Univariate: an equation containing
only one variable
( e.g. Unary )
Multivariate: an equation containing
more than one variable
( e.g. n-ary )
Mathematics
Expression constituents consisting of Constants and
Variables exhibiting exclusive parity
Polynomial

6. © Art Traynor 2011
Mathematics
Definitions
Expression
Symbol Operation Relation
Expression – A Mathematical Sentence
Proposition Formula
VariablesConstants
Operands ( Terms )
Equation
A formula stating an
equivalency class relation
Linear Equation
An equation in which each term is either
a constant or the product of a constant
and (a) variable[s] of the first degree
Mathematics
Polynomial

7. © Art Traynor 2011
Mathematics
Expression
Mathematical Expression
A representational precursive discrete composition to a
Mathematical Statement or Proposition ( e.g. Equation )
consisting of :

Operands / Terms
Expression
A well-formed symbolic
representation of Operands
( Terms or Monomials ) ,
of discrete arity, upon which one
or more Operations ( LOC’s ) may
structure a Relation
Mathematics
n Scalar Constants ( i.e. Coefficients )
n Variables or Unknowns
The Cardinality of which is referred to as the Arity of the Expression
Constituent representational Symbols composed of :
Algebra
Laws of Composition ( LOC’s )
Governs the partition of the Expression
into well-formed Operands or Terms
( the Cardinality of which is a multiple of Monomials )

8. © Art Traynor 2011
Mathematics
Arity
Arity
Expression
The enumeration of discrete symbolic elements ( Variables )
comprising a Mathematical Expression
is defined as its Arity

The Arity of an Expression can be represented by
a non-negative integer index variable ( ℤ + or ℕ ),
conventionally “ n ”

A Constant ( Airty n = 0 , index ℕ )or Nullary
represents a term that accepts no Argument

A Unary expresses an Airty n = 1
A relation can not be defined for
Expressions of Arity less than
two: n < 2
A Binary expresses Airty n = 2
All expressions possessing Airty n > 1 are n-ary, Multary, Multiary, or Polyadic
VariablesConstants
Operands
Expression
Polynomial

9. © Art Traynor 2011
Mathematics
Expression
Arity
Operand
 Arithmetic : a + b = c
The distinct elements of an Expression
by which the structuring Laws of Composition ( LOC’s )
partition the Expression into discrete Monomial Terms
 “ a ” and “ b ” are Operands
 The number of Variables of an Expression is known as its Arity
n Nullary = no Variables ( a Scalar Constant )
n Unary = one Variable
n Binary = two Variables
n Ternary = three Variables…etc.
VariablesConstants
Operands
Expression
Polynomial
n “ c ” represents a Solution ( i.e. the Sum of the Expression )
Arity is canonically
delineated by a Latin
Distributive Number,
ending in the suffix “ –ary ”

10. © Art Traynor 2011
Mathematics
Arity
Arity ( Cardinality of Expression Variables )
Expression
A relation can not be defined for
Expressions of Arity less than
two: n < 2
Nullary
Unary
n = 0
n = 1
Binary n = 2
Ternary n = 3
1-ary
2-ary
3-ary
Quaternary n = 4 4-ary
Quinary n = 5 5-ary
Senary n = 6 6-ary
Septenary n = 7 7-ary
Octary n = 8 8-ary
Nonary n = 9 9-ary
n-ary
VariablesConstants
Operands
Expression
Polynomial
0-ary

11. © Art Traynor 2011
Mathematics
Operand
Parity – Property of Operands
Parity
n is even if ∃ k n = 2k
n is odd if ∃ k n = 2k+1
Even ↔ Even
Integer Parity
Same Parity
Even ↮ Odd Opposite Parity
|:
|:

12. © Art Traynor 2011
Mathematics
Polynomial
Expression
A well-formed symbolic
representation of operands, of
discrete arity, upon which one
or more operations can
structure a Relation
Expression
Polynomial Expression
A Mathematical Expression ,
the Terms ( Operands ) of which are a compound composition of :
Polynomial
Constants – referred to as Coefficients
Variables – also referred to as Unknowns
And structured by the Polynomial Structure Criteria ( PSC )
arithmetic Laws of Composition ( LOC’s ) including :
Addition / Subtraction
Multiplication / Non-Negative Exponentiation
LOC ( Pn ) = { + , – , x bn ∀ n ≥ 0 }
Wiki: “ Polynomial ”
An excluded equation by
Polynomial Structure Criteria ( PSC )
Σ an xi
n
i = 0
P( x ) = an xn + an – 1 xn – 1 +…+ ak+1 xk+1 + ak xk +…+ a1 x1 + a0 x0
Variable
Coefficient
Polynomial Term
From the Greek Poly meaning many,
and the Latin Nomen for name





13. © Art Traynor 2011
Mathematics
Degree
Expression
Polynomial
Degree of a Polynomial
Polynomial
Wiki: “ Degree of a Polynomial ”
The Degree of a Polynomial Expression ( PE ) is supplied by that
of its Terms ( Operands ) featuring the greatest Exponentiation
For a multivariate term PE , the Degree of the PE is supplied by that
Term featuring the greatest summation of Variable exponents

P = Variable Cardinality & Variable Product
Exponent Summation
& Term Cardinality
Arity
Latin “ Distributive ” Number
suffix of “ – ary ”
Degree
Latin “ Ordinal ” Number
suffix of “ – ic ”
Latin “ Distributive ” Number
suffix of “ – nomial ”
0 =
1 =
2 =
3 =
Nullary
Unary
Binary
Tenary
Constant
Linear
Quadratic
Cubic
Monomial
Binomial
Trinomial
An Expression composed of
Constants ( Coefficients ) and
Variables ( Unknowns) with an
LOC of Addition, Subtraction,
Multiplication and Non-
Negative Exponentiation

14. © Art Traynor 2011
Mathematics
Degree
Polynomial
Degree of a Polynomial
Nullary
Unary
p = 0
p = 1 Linear
Binaryp = 2 Quadratic
Ternaryp = 3 Cubic
1-ary
2-ary
3-ary
Quaternaryp = 4 Quartic4-ary
Quinaryp = 5 5-ary
Senaryp = 6 6-ary
Septenaryp = 7 7-ary
Octaryp = 8 8-ary
Nonaryp = 9 9-ary
“ n ”-ary
Arity Degree
Monomial
Binomial
Trinomial
Quadranomial
Terms
Constant
Quintic
P
Wiki: “ Degree of a Polynomial ”
Septic
Octic
Nonic
Decic
Sextic
aka: Heptic
aka: Hexic

15. © Art Traynor 2011
Mathematics
Degree
Expression
Polynomial
Degree of a Polynomial
Polynomial
Wiki: “ Degree of a Polynomial ”
An Expression composed of
Constants ( Coefficients ) and
Variables ( Unknowns) with an
LOC of Addition, Subtraction,
Multiplication and Non-
Negative Exponentiation
The Degree of a Polynomial Expression ( PE ) is supplied by that
of its Terms ( Operands ) featuring the greatest Exponentiation
For a PE with multivariate term(s) ,
the Degree of the PE is supplied by
that Term featuring the greatest summation
of individual Variable exponents

P( x ) = ai xi
0 Nullary Constant Monomial
P( x ) = ai xi
1
Unary Linear Monomial
P( x ) = ai xi
2
Unary Quadratic Monomial
ai xi
1 yi
1P( x , y ) =
Binary Quadratic Monomial
Univariate
Bivariate

16. © Art Traynor 2011
Mathematics
Degree
Expression
Polynomial
Degree of a Polynomial
Polynomial
Wiki: “ Degree of a Polynomial ”
The Degree of a Polynomial Expression ( PE ) is supplied by that
of its Terms ( Operands ) featuring the greatest Exponentiation
For a multivariate term PE , the Degree of the PE is supplied by that
Term featuring the greatest summation of Variable exponents

P( x ) = ai xi
0 Nullary Constant Monomial
P( x ) = ai xi
1
Unary Linear Monomial
P( x ) = ai xi
2
Unary Quadratic Monomial
ai xi
1 yi
1P( x , y ) = Binary Quadratic Monomial
ai xi
1 yi
1zi
1P( x , y , z ) = Ternary Cubic Monomial
Univariate
Bivariate
Trivariate
Multivariate

17. © Art Traynor 2011
Mathematics
Quadratic
Expression
Polynomial
Quadratic Polynomial
Polynomial
Wiki: “ Degree of a Polynomial ”
A Unary or greater Polynomial
composed of at least one Term and :
Degree precisely equal to two
Quadratic ai xi
n ∀ n = 2
 ai xi
n yj
m ∀ n , m n + m = 2|:
Etymology
From the Latin “ quadrātum ” or “ square ” referring
specifically to the four sides of the geometric figure
Wiki: “ Quadratic Function ”
Arity ≥ 1
 ai xi
n ± ai + 1 xi + 1
n ∀ n = 2
Unary Quadratic Monomial
Binary Quadratic Monomial
Unary Quadratic Binomial
 ai xi
n yj
m ± ai + 1 xi + 1
n ∀ n + m = 2 Binary Quadratic Binomial

18. © Art Traynor 2011
Mathematics
Equation
Equation
Expression
An Equation is a statement or Proposition
( aka Formula ) purporting to express
an equivalency relation between two Expressions :

Expression
Proposition
A declarative expression
asserting a fact whose truth
value can be ascertained
Equation
A symbolic formula, in the form of a
proposition, expressing an equality relationship
Formula
A concise symbolic
expression positing a
relationship between
quantities
VariablesConstants
Operands
Symbols
Operations
The Equation is composed of
Operand terms and one or more
discrete Transformations ( Operations )
which can render the statement true
( i.e. a Solution )
Polynomial

19. © Art Traynor 2011
Mathematics
Equation
Solution
Solution and Solution Sets
 Free Variable: A symbol within an expression specifying where
a substitution may be made
Contrasted with a Bound Variable
which can only assume a specific
value or range of values
 Solution: A value when substituted for a free variable which
renders an equation true
Analogous to independent &
dependent variables
Unique Solution: only one solution
can render the equation true
(quantified by $! )
General Solution: constants are
undetermined
General Solution: constants are
value-specified (bound?)
Unique Solution
Particular Solution
General Solution
Solution Set
n A family (set) of all solutions –
can be represented by a parameter (i.e. parametric representation)
 Equivalent Equations: Two (or more) systems of equations sharing
the same solution set
Section 1.1, (Pg. 3)
Section 1.1, (Pg. 3)
Section 1.1, (Pg. 6)
Any of which could include a Trivial Solution
Section 1.2, (Pg. 21)

20. © Art Traynor 2011
Mathematics
Equation
Solution
Solution and Solution Sets
 Solution: A value when substituted for a free variable which
renders an equation true
Unique Solution: only one solution
can render the equation true
(quantified by $! )
General Solution: constants are
undetermined
General Solution: constants are
value-specified (bound?)
Solution Set
n For some function f with parameter c such that
f(xi , xi+1 ,…xn – 1 , xn ) = c
the family (set) of all solutions is defined to include
all members of the inverse image set such that
f(x) = c ↔ f -1(c) = x
f -1(c) = {(ai , ai+1 ,…an-1 , an ) Ti· Ti+1 ·…· Tn-1· Tn |f(ai , ai+1 ,…an-1 , an ) = c }
where Ti· Ti+1 ·…· Tn-1· Tn is the domain of the function f
o f -1(c) = { }, or Ø empty set ( no solution exists )
o f -1(c) = 1, exactly one solution exists ( Unique Solution, Singleton)
o f -1(c) = { cn } , a finite set of solutions exist
o f -1(c) = {∞ } , an infinite set of solutions exists
Inconsistent
Consistent
Section 1.1,
(Pg. 5)

21. © Art Traynor 2011
Mathematics
Linear Equation
Linear Equation
Equation
An Equation consisting of:
Operands that are either
Any Variables are restricted to the First Order n = 1
Linear Equation
An equation in which each term
is either a constant or the
product of a constant and (a)
variable[s] of the first order
Expression
Proposition
Equation
Formula
n Constant(s) or
n A product of Constant(s) and
one or more Variable(s)
The Linear character of the Equation derives from the
geometry of its graph which is a line in the R2 plane

As a Relation the Arity of a Linear Equation must be
at least two, or n ≥ 2 , or a Binomial or greater Polynomial

Polynomial

22. © Art Traynor 2011
Mathematics
Equation
Linear Equation
Linear Equation
 An equation in which each term is either a constant or the product
of a constant and (a) variable[s] of the first order
Term ai represents a Coefficient
b = Σi= 1
n
ai xi = ai xi + ai+1 xi+1…+ an – 1 xn – 1 + an xn
Equation of a Line in n-variables
 A linear equation in “ n ” variables, xi + xi+1 …+ xn-1 + xn
has the form:
n Coefficients are distributed over a defined field
(e.g. ℕ , ℤ , ℚ , ℝ , ℂ )
Term xi represents a Variable ( e.g. x, y, z )
n Term a1 is defined as the Leading Coefficient
n Term x1 is defined as the Leading Variable
Section 1.1, (Pg. 2)
Section 1.1, (Pg. 2)
Section 1.1, (Pg. 2)
Section 1.1, (Pg. 2)
Coefficient = a multiplicative factor
(scalar) of fixed value (constant)
Section 1.1, (Pg. 2)

23. © Art Traynor 2011
Mathematics
Linear Equation
Equation
Standard Form ( Polynomial )
 Ax + By = C
 Ax1 + By1 = C
For the equation to describe a line ( no curvature )
the variable indices must equal one

 ai xi + ai+1 xi+1 …+ an – 1 xn –1 + an xn = b
 ai xi
1 + ai+1 x 1 …+ an – 1 x 1 + a1 x 1 = bi+1 n – 1 n n
ℝ
2
: a1 x + a2 y = b
ℝ
3
: a1 x + a2 y + a3 z = b
Blitzer, Section 3.2, (Pg. 226)
Section 1.1, (Pg. 2)
Test for Linearity
 A Linear Equation can be expressed in Standard Form
As a species of Polynomial , a Linear Equation
can be expressed in Standard Form
 Every Variable term must be of precise order n = 1
Linear Equation
An equation in which each term
is either a constant or the
product of a constant and (a)
variable[s] of the first order
Expression
Proposition
Equation
Formula
Polynomial

24. © Art Traynor 2011
Mathematics
Discrete Structures
Set ( Naïve )
 A Set is an collection of Mathematical Objects, minimally structured
by a membership Relation , and lacking Order.
Sets
Relation
A Set is Structured by some
characteristic Relation
By this membership definitional Relation ( e.g. Ф )
a Set is thus understood to Contain its Elements

Inclusion
 As the minimal structure defining a Set,
Set membership or Inclusion constitutes a Proposition
for which a truth value can be ascertained
Lay, Section 2.5,
(Pg. 32)
O’Leary, Section 1.1,
Pg. 3
Inclusion is asserted as a argument to a Logical Predicate
Those Set elements which can be substituted for “ a ” evaluating
to a true statement form the Domain of the Predicate
O’Leary, Section 2.1,
Pgs. 41 & 46
A = { a | Ф( a ) }

25. © Art Traynor 2011
Mathematics
Set
Inclusion
Relation
A Set is Structured by some
characteristic Relation
Inclusion
 As the minimal structure defining a Set,
Set membership or Inclusion constitutes a Proposition
for which a truth value can be ascertained
Lay, Section 2.5,
(Pg. 32)
A = { a | Ф( a ) }
O’Leary, Section 1.1,
Pg. 3
Inclusion is asserted as a argument
to a Logical Predicate ( LP ) e.g. Ф( a )

Those Set elements which can be substituted for “ a ” evaluating
to a true statement form the Domain of the Predicate
O’Leary, Section 2.1,
Pgs. 41 & 46
n The LP completely describes the elements “ a ” of the Set “ A ”
so for “ a ” to be an Element of “ A ” , Ф( a ) must be True
n An inclusion argument must evaluate to True or False, not both
n Conversely if Ф( a ) is True then “ a ” is in “ A ”
a ∈ A iff Ф( a )
O’Leary, Section 3.1,
Pgs. 97

26. © Art Traynor 2011
Mathematics
Relation
A Set is Structured by some
characteristic Relation
Inclusion
 As the minimal structure defining a Set,
Set membership or Inclusion constitutes a Proposition
for which a truth value can be ascertained
Lay, Section 2.5,
(Pg. 32)
A = { a | Ф( a ) }
O’Leary, Section 1.1,
Pg. 3
Inclusion is asserted as a argument
to a Logical Predicate ( LP ) e.g. Ф( a )

n If any “ a ” will satisfy Ф i.e. rendering Ф( a ) true
then the Image of Ф shares the same Cardinality as its
Codomain which is the Universe “ U ” of all points
U
A
U = A
n “ U ” and “ A ” thus form a conjunction of Improper Subsets
A ⊆ U ∧ U ⊆ A
Also known as
Property of Antisymmetric Equality of Sets ( POAEOS )
O’Leary, Section 3.3,
Pg. 114
Wiki: “ Algebra of Sets ”
Set
Inclusion

27. © Art Traynor 2011
Mathematics
Cardinality
Set Inclusion may be reckoned
as a continuum of comparative Cardinality
Exhaustive
Inclusion
Inexhaustive
Inclusion
Equality
Improper Inclusion
Proper Inclusion
Set
Cardinality
Continuum
Equality
Exhaustive Inclusion
Ø ≠ ( A = B ) ⟺ ( A ⊆ B ∧ B ⊆ A )
U
A = B
Ø ≠ ( A = B ) ⟺ { x ( ∀ x ) ( x ∈ A ∧ x ∈ B )|: O’Leary, Section 3.3,
Pg. 114
Set
Inclusion

28. © Art Traynor 2011
Mathematics
Cardinality
Set Inclusion may be reckoned
as a continuum of comparative Cardinality
Exhaustive
Inclusion
Inexhaustive
Inclusion
Equality
Improper Inclusion
Proper Inclusion
Set
Cardinality
Continuum
Equality
Exhaustive Inclusion
U
A = B
O’Leary, Section 3.3,
Pg. 114
State of the Uniono
Ø ≠ A = B = { x ( ∀ x ) ( x ∈ A ∧ x ∈ B ) }|:
Ø ≠ { A ∪ B } = { x ( ∃ x ) ( x ∈ A ∨ x ∈ B ) }|:
Set
Inclusion

29. © Art Traynor 2011
Mathematics
Cardinality
Set Inclusion may be reckoned
as a continuum of comparative Cardinality
Exhaustive
Inclusion
Inexhaustive
Inclusion
Equality
Improper Inclusion
Proper Inclusion
Set
Cardinality
Continuum
Equality
Exhaustive Inclusion
U
A = B
O’Leary, Section 3.3,
Pg. 114
Intersection Congestiono
Ø ≠ A = B = { x ( ∀ x ) ( x ∈ A ∧ x ∈ B ) }|:
Ø ≠ { A ∩ B } = { x ( ∃ x ) ( x ∈ A ∧ x ∈ B ) }|:
Set
Inclusion

30. © Art Traynor 2011
Mathematics
Cardinality
Set Inclusion may be reckoned
as a continuum of comparative Cardinality
Exhaustive
Inclusion
Inexhaustive
Inclusion
Equality
Improper Inclusion
Proper Inclusion
Set
Cardinality
Continuum
Equality
Exhaustive Inclusion
U
A = B
U
A ≤ B
Improper Subset
Quasi-Complete Inclusion
Ø ≠ ( A ⊆ B ) ⟺ { x ( ∀ x ) ( x ∈ A ⇒ x ∈ B )|:
O’Leary, Section 3.2,
Pg. 109
Set
Inclusion

31. © Art Traynor 2011
Mathematics
Cardinality
Set Inclusion may be reckoned
as a continuum of comparative Cardinality
Exhaustive
Inclusion
Inexhaustive
Inclusion
Equality
Improper Inclusion
Proper Inclusion
Set
Cardinality
Continuum
Equality
Exhaustive Inclusion
U
A = B
U
A ≤ B
Improper Subset
Quasi-Complete Inclusion O’Leary, Section 3.2,
Pg. 109
Posits a Logical “ OR ”o
Equality is a possibilityo
Set
Inclusion

32. © Art Traynor 2011
Mathematics
Cardinality
Set Inclusion may be reckoned
as a continuum of comparative Cardinality
Exhaustive
Inclusion
Inexhaustive
Inclusion
Equality
Improper Inclusion
Proper Inclusion
Set
Cardinality
Continuum
Equality
Exhaustive Inclusion
U
A = B
U
A ≤ B
Improper Subset
Quasi-Complete Inclusion O’Leary, Section 3.2,
Pg. 109
State of the Uniono
Ø ≠ { A ∪ B } = { x ( ∀ x ) ( x ∈ A ∨ x ∈ B ) }|:
Ø ≠ { A ∪ B } = { x ¬ ( ∀ x ) ( x ∈ A ⇒ x ∈ B ) }
Equality
Or
Ø ≠ { A ∪ B } = { x ( ∃ x ) ( x ∈ A ∧ x ∉ B) }
|:
|:
Inequality
Set
Inclusion

33. © Art Traynor 2011
Mathematics
Cardinality
Set Inclusion may be reckoned as a continuum of comparative Cardinality
Exhaustive
Inclusion
Inexhaustive
Inclusion
Equality
Improper Inclusion
Proper Inclusion
Set
Cardinality
Continuum
Equality
Exhaustive Inclusion
U
A = B
U
A ≤ B
Improper Subset
Quasi-Complete Inclusion O’Leary, Section 3.2,
Pg. 109
Intersection Congestiono
Ø ≠ { A ∩ B } = { x ( ∀ x ) ( x ∈ A ∧ x ∈ B ) }|:
Ø ≠ { A ∩ B } = { x ( ∀ x ) ( x ∈ A ⇒ x ∈ B ) }
Equality
Or
Ø ≠ { A ∩ B } = { x ¬ ( ∀ x ) ( x ∈ A ∧ x ∉ B) }
|:
|:Inequality
Ø ≠ { A ∩ B } = { x ( ∃ x ) ( x ∈ A ⇒ x ∈ B) }|:
Non-Empty
Minimal?
Set
Inclusion

34. © Art Traynor 2011
Mathematics
Cardinality
Set Inclusion may be reckoned
as a continuum of comparative Cardinality
Exhaustive
Inclusion
Inexhaustive
Inclusion
Equality
Improper Inclusion
Proper Inclusion
Set
Cardinality
Continuum
Equality
Exhaustive Inclusion
U
A = B
U
A ≤ B
Improper Subset
Quasi-Complete Inclusion
Ø ≠ ( A ⊂ B ) ⟺ { x ( ∃ x ) ( x ∈ A ∧ x ∉ B )|:
O’Leary, Section 3.2,
Pg. 110
U
B
Proper Subset
Inexhaustive Inclusion
Ø ≠ ( A ⊂ B ) ⟺ { x ¬ ( ∀ x ) ( x ∈ A ⇒ x ∈ B )|:
A
Set
Inclusion

35. © Art Traynor 2011
Mathematics
Cardinality
Set Inclusion may be reckoned
as a continuum of comparative Cardinality
Exhaustive
Inclusion
Inexhaustive
Inclusion
Equality
Improper Inclusion
Proper Inclusion
Set
Cardinality
Continuum
Equality
Exhaustive Inclusion
U
A = B
U
A ≤ B
Improper Subset
Quasi-Complete Inclusion
Excludes or Forbids Equality
O’Leary, Section 3.2,
Pg. 110
U
B
Proper Subset
Inexhaustive Inclusion
A
o
Set
Inclusion

36. © Art Traynor 2011
Mathematics
Relation
A Set is Structured by some
characteristic Relation
Equality
 Equality of Sets can be regarded as an Exhaustive Inclusion
Logical Equality
U
A
U = A
Wiki: “ Equality
(Mathematics) ”
∀ ( a , b ) , ( a = b ) ⇒ ( Ф( a ) ⟺ Ф( b ) )
Two distinct symbols which refer to the same object
n Logical Predicate ( LP ) Interpretation of Equality
Set Theoretic Equality
Two distinct Sets share the same Cardinality
Algebraic Equality
Two distinct Expressions evaluating to the same value
n An Expression which posits an Equivalency Class Relation
|A | = |B |
n Satisfies the Reflexive , Symmetric , and Transitive Properties
Set
Equality

37. © Art Traynor 2011
Mathematics
Cardinality
The Exclusion of select Elements of a Set
may be necessary
to invest the restricted Set
with added Structure
that an unrestricted Inclusion might otherwise prohibit
 Singleton Exclusion
a ∉ B
A species of logical “ not ” or negative inclusion of a single element from a Set
Examples:
Closure preservation of a Set , Field , or Ring
under a Law of Composition ( LOC ) such as Multiplication
where division by zero , or a zero Inverse inclusion,
would map an element outside of the defining construct
Set
Exclusion

38. © Art Traynor 2011
Mathematics
Cardinality
The Exclusion of select Elements of a Set
may be necessary
to invest the restricted Set
with added Structure
that an unrestricted inclusion might otherwise prohibit
 Singleton Exclusion
 Roster Exclusion
a , c ∉ B
A species of logical “ not ” or negative inclusion of multiple, discrete elements from a Set
Set
Exclusion

39. © Art Traynor 2011
Mathematics
Cardinality
The Exclusion of select Elements of a Set
may be necessary
to invest the restricted Set
with added Structure
that an unrestricted inclusion might otherwise prohibit
 Singleton Exclusion
 Roster Exclusion
 Complement
U
A
Complement
A (shaded)
A form of Universal Quantification ( i.e. Exhaustive Exclusion )
whereby the Complement of a Set
is populated by all those elements not specifically included
in the referent Set
 A = { A shaded)
Set
Exclusion

40. © Art Traynor 2011
Mathematics
Cardinality
The Exclusion of select Elements of a Set
may be necessary
to invest the restricted Set
with added Structure
that an unrestricted inclusion might otherwise prohibit U
A
Complement
A (shaded)
 Complement Exclusion
a ∉ p ·ℤ
Set
Exclusion
 Difference Exclusion
A species of
Binary Exclusion
A B = { a a A ∧ a B }|:
U
A
Difference
A B ∧ B A
B
a b
B A = { b b A ∧ b B }|:
A species of
Logical “ Not ”
O’Leary, Section 3.1,
Pg. 101
A = U A = { x x  U ∧ x A }|:
x
O’Leary, Section 3.1,
Pg. 102
If { A ∩ B } ≠ Ø then the Sets are
considered Disjoint or Mutually Exclusive

41. © Art Traynor 2011
Mathematics
Fundamentals
U
A B
A  B
Intersection
U
A B
A  B
Union
U
A B
Difference
B
A – B = A  B
U
A
Complement
A (shaded)
Set Difference: The set containing
those elements that are in A but not in B
x belongs to the difference of A and B
if-and-only x A and x B
A – B = {x |x A  x B }
Set
Exclusion

42. © Art Traynor 2011
Mathematics
Sets
Subsets
Subsets
The set A is a subset of the set B
if and only if
every element of Ais also an element of B
A B
≤
The SUBset
Lesser or Equal
Cardinality
Greater or Equal
Cardinality
The SUPERset
Notation functions like the
“less-than-or-equal-to”
symbol
"x ( x A  x B )
Every non-empty set S has at least two subsets
the empty set 
and the set S itself
"x ( x A  x B ) $x ( x B  x A )
 S
S S
If set A is a subset of the set B
and A ≠ B
then A is a proper subset of B A B
such that A B
and there is at least one element of A not also in B
Rosen, Section 2.1,
(Pg. 119)
A B
The SUBset The SUPERset
<
Notation functions like the
“less-than” symbol
Rosen, Section 2.1,
(Pg. 120)

43. © Art Traynor 2011
Mathematics
Sets
Union of Sets
Union of Sets Section 2.2, (Pg. 127)
Given A and Bare sets
there exists a collection of their unique elements
forming a distinct set A B
among whose subsets A and B are constituent elements

"n F |A B =(n A ) ( n  B )
A B =(A  ( A B ))  (B  ( A B ))
P ( A B ) = { , A  B , A, B }
A union need not contain
any shared elements of
the constituent sets
If the constituent sets
contain common elements
those elements are only
included in the union once
Examples:
A = { 1, 3, 5 }
B = { 2, 4, 6 }
A B = {1, 2, 3, 4, 5, 6 }
A = { 1, 2, 5 }
B = { 2, 4, 6 }
A B = {1, 2, 4, 5, 6 }
|A | = 1
A = { 2, 2, 2 }
Equivalent elements of a set
do not affect its cardinality
* Explore the implications of
Professor Francis Su’s
( Harvey Mudd College
course in Real Analysis )
assertion of the logical
equivalence of union to “OR”

44. © Art Traynor 2011
Mathematics
Sets
Union of Sets
Union of Sets Section 2.2, (Pg. 127)
Given A and Bare sets
there exists a collection of their unique elements
forming a distinct set A B
among whose subsets A and B are constituent elements
 * Explore the implications of
Professor Francis Su’s
( Harvey Mudd College
course in Real Analysis )
assertion of the logical
equivalence of union to “OR”
A B = { x | x  A  x  B }

45. © Art Traynor 2011
Mathematics
Sets
Intersection of Sets
Intersection of Sets Section 2.2, (Pg. 127)
Given A and Bare sets
and sets A and B contain at least one element in common,
there exists a collection of their unique elements
forming a distinct set A B
among whose elements are only those in common to both

"n F |A B =(n A ) ( n  B )
A B =(A  ( A B ))  (B  ( A B ))
P ( A B ) = { , A  B , A, B }

46. © Art Traynor 2011
Mathematics
Subsets
Given set S , the power set of S
is the set of all subsets of S
denoted P( S )
Given set S = { , S, a1 , a2 ,…, an }
|P( S )| = 2n
Power Set
Sets
Subsets
Rosen 2.1, (Pg. 121)

47. © Art Traynor 2011
Mathematics
Set Operations
For sets A and B
the cartesian product of A and B
denoted Ax B
Cartesian Product ( CP )
is the set of all ordered pairs ( a , b )
where a A and b B
A x B = {(a, b ) |a A  b B }
A = { 1, 2 } B= { a, b, c }
A x B = {(1, a ), ( 1, b ), ( 1, c ),
( 2, a ), ( 2, b ), ( 2, c ) }
Sets
Operations
Rosen 2.1, (Pg. 123)
Sets are unordered
A CP of Sets generates a product set of ordered n-tuples

48. © Art Traynor 2011
Mathematics
Discrete Structures
Fundamentals
Sets
U
A B
A  B
Intersection
U
A B
A  B
Union
U
A B
Difference
B
A – B = A  B
U
A
Complement
A (shaded)
Set Difference: The set containing
those elements that are in A but not in B
x belongs to the difference of A and B
if-and-only x A and x B
A – B = {x |x A  x B }

49. © Art Traynor 2011
Mathematics
Relations
A binary relation from set A to set B
is a set R of ordered pairs ( a, b )
where a A and b B ( a, b ) R
Binary Relation
denoted aR b
a is related to b by R
Sets
Subsets

50. © Art Traynor 2011
Mathematics
Functions
Relational Taxonomy
A Function is a relation between a set of inputs ( Domain / Argument
/ Independent Variable ) and outputs ( Codomain / Value / Dependent
Variable ) such that each input is associated with only a single output
( akin to a Cartesian Product, yielding an Ordered Pair )

Bijective
Functions
Injective
Functions
Surjective
Functions
Discrete Structures
Function
Relations
A Relation is a structuring of a Set to which a truth value can be assigned
Structure increases as the
heirachy graduates
Decreasing
Structure
Increasing
Structure

51. © Art Traynor 2011
Mathematics
Discrete Structures
Rules of Inference
Quantified Statements
"x P(x )
P( c )
Universal Instantiation
P( c ) for an arbitrary “ c ”
"x P(x )
Universal Generalization
$x P( x )
P( c ) for some element “ c ”
Existential Instantiation
P( c ) for some element “ c ”
$x P( x )
Existential Generalization

52. © Art Traynor 2011
Mathematics
Discrete Structures
Mathematical Induction
Proof
It is not assumed that P(k) is true for Z+
only that if it is assumed P(k) is true,
then P(k+1) is also true

Well -Ordering Property ( WOP ) of Z+
Every non-empty subset of Z+
has a least element
“that” something is the case….
purports to demonstrate a conclusion
on the basis of evidence
If we assume P(1) is true,
and further that P(k)  P(k + 1) is true for n Z+
Assume there is at least one n Z+ for which P(n) is false


 Then {S}Z+ |P(n) is false ≠ 
 Then {S}has a “least element” m {S}Z+ |P(n) is false
“The WOPper”
m ≠ 1, because P(1) is true, which is a contradiction

53. © Art Traynor 2011
Mathematics
Discrete Structures
Sets
Number Sets ( Fields )
N
Z
= { 0, 1, 2, 3, … } the set of Natural Numbers
= { 0, 1, 2, 3, … } the set of Integers (after Zahlen, German for “ numbers ” )
Z+
= { 0, 1, 2, 3, … } the set of Positive Integers
Q = { 0, 1, 2, 3, … } the set of Rational Numbers, after “ Q ” for quotient
Has manifold useful properties
 Countability is established with the showing of a one-to-one correspondence
 Cardinality: the littlest giant, |Z+ | = 0

54. © Art Traynor 2011
Mathematics
Relations
A binary relation from set A to set B
is a set R of ordered pairs ( a, b )
where a A and b B ( a, b ) R
Binary Relation
denoted aR b
a is related to b by R
Sets
Subsets

55. © Art Traynor 2011
Mathematics
Properties of Relations
A relation R on a set A is reflexive
If ( a, a )  R and for every element a A
Reflexive Property
denoted aR a
"a (( a, a )  R )
A relation R on a set A is symmetric
if ( b, a )  R whenever ( a, b )  R for all a, b A
Symmetric Property
"a "b (( a, b ) R  ( b, a ) R )
A relation is symmetric if and only if “ a is related to b ” implies “ b is related to a ”
i.e. ( b, a ) belongs to the relation whenever ( a, b ) does
a = b implies b = a and that the relation is symmetric
Sets
Subsets

56. © Art Traynor 2011
Mathematics
Properties of Relations
A relation R on a set A
such that for all ( a, b )  A if
Antisymmetric Property
"a "b ((( a, b ) R  ( b, a ) R )  ( a = b ))
A relation is antisymmetric if and only if there are no distinct pairs ( a, b )
where a is related to b, AND b is related to a
that if ( a, b ) R , a must equal b
( a, b )  R and
( b, a )  R then
a = b is antisymmetric
i.e. a = b is the only permitted relationship that antisymmetry will admit
where ( a, b ) R and ( b, a ) R


EXCEPT
Sets
Subsets

57. © Art Traynor 2011
Mathematics
Properties of Relations
A relation R on a set A is transitive
If whenever ( a, b )  R and
Transitive Property
( b, c )  R then
( a, c )  R for all a, b, c  R
"a "b "c ((( a, b ) R  ( b, c ) R )  ( a, c ) R )
A relation on a set A is an equivalence relation if it is
Equivalence Relation
reflexive, symmetric, transitive
Sets
Subsets

58. © Art Traynor 2011
Mathematics
Discrete Structures
Set Identities
A U = A
Identity Laws
Domination Laws
Idempotent Laws
Sets
A  = A
A U = U
A  = 
A A = A
A A= A
Complement Laws
A A = U
A A= 
Complement Law( A ) = A

59. © Art Traynor 2011
Mathematics
Discrete Structures
Set Identities
A B = B  A
Commutative Laws
Associative Laws
Distributive Laws
Sets
A  B = B  A
A ( B C ) = ( A  B ) C
Absorption Laws
De Morgan’s Law
A ( B C ) = ( A  B ) C
A ( B C ) = ( A  B )  ( A  C )
A ( B C ) = ( A  B )  ( A  C )
The Union of the Intersection is equal
to the intersection of the unions
The Intersection of the Union is equal
to the union of the intersections
A ( A B ) = A
A ( A B ) = A
A  B = A  B
A B = A  B
The Union of a set’s Intersection
(with another set) is an Identity
The Intersection of a set’s Union
(with another set) is an Identity

60. © Art Traynor 2011
Mathematics
Discrete Structures
Sequence Collection Notation
 Union:  Ai = A1  A2 … An
i =1
Sets
n
Lay, Section 2.5, (Pg. 39)

61. © Art Traynor 2011
Mathematics
Discrete Structures
Usage Conventions for SBN
 A Set is a collection of Mathematical Objects, minimally structured by a
membership Relation, without need of any particular Order.
Set Builder Notation
{ x | Ф(x) }
( Logical ) Predicate
( Predicate ) Extension or Set
Conditional Separator
Argument
Argument
Can be generalized to constitute
a “ Term ” or “ Expression ”
Predicate
Populated with a propositional
expression, logically equivalent
to a Formula, capable of
quantification, and that can be
evaluated to a truth value
The SBN Extension presupposes
some Universe of Discourse over
which satisfaction of the Predicate
constitutes membership in the Set
U
Universe of Discourse
Extension ( Set )
Those elements which satisfy the
Predicate constitute elements of
the Set, or the Extension of the
Predicate ( i.e. Image Set )
Intension
Intension
An Well-Formed expression
composed of a quantified
Argument and Predicate
sufficient to evaluate to a truth
value ( i.e. Domain Set )
A well-formed Set can be constructed of the following constituents:
{ x | Ф(x) }

62. © Art Traynor 2011
Mathematics
Discrete Structures
Usage Conventions for SBN
 A Set is a collection of Mathematical Objects, minimally structured by a
membership Relation, without need of any particular Order.
Set Builder Notation
The Set Extension Representation ( SER ) is adequate to denote the
Set, yet can be rendered more resilient ( less ambiguous ) by added
specification
S = { x | Ф(x) }
If you’re going to have a Set, you
may as well name it…
Set Membership can be explicitly denoted by the symbol “ ” which
assigns the Relation “ is an element of ” to members of the Set

The SER ( aka : the Predicate Extension ) can be uniquely denoted
by assigning to it a Variable designation:

S = { x | x ℝ , Ф(x) }
U
Universe of Discourse
S

63. © Art Traynor 2011
Mathematics
Discrete Structures
Usage Conventions for SBN
Set Builder Notation
S = { x | x ℝ , Ф(x) } → { ℝ }
( Logical ) Predicate
Conditional Separator
Argument
Argument
Can be generalized to constitute
a “ Term ” or “ Expression ”
Predicate (Bifurcated)
Compound expression composed
of a quantification clarifying a Field
Membership Relation, juxtaposed
by comma delineation, with a
Function Image Map
Domain Quantification
Ratifies Field Membership Relation
implied universal instantiation ( IUI )
Set Name (Variable)
 A Set is a collection of Mathematical Objects, minimally structured by a
membership Relation, without need of any particular Order.
The Set Extension Representation ( SER ) is adequate to denote the
Set, yet can be rendered more resilient ( less ambiguous ) by added
specification
Domain Quantification
Image Map Image Map
A Formula (e.g. Function),
capable of quantification, which
can be evaluated to a truth value
Declarative Delineator
Image Set
> Named Set
> Single Free Variable - IUI
( Implicit Universal Instantiation )
> Predicate Ratified Field
Membership ( PRFM )
> Compound Predicate,
Comma Delineated
( Implicit Conjunctive )
> Function Image Map Specification
> Roster Image Set

64. © Art Traynor 2011
Mathematics
Discrete Structures
Usage Conventions for SBN
Set Builder Notation
S = { x | x ℝ  x = x2 } → { 0 , 1 }
( Logical ) Predicate
Conditional Separator
Argument
Argument
Can be generalized to constitute
a “ Term ” or “ Expression ”
Predicate (Bifurcated)
Compound expression composed of
a quantification clarifying a Field
Membership Relation with
conjunctive image map specification
Domain Quantification
Ratifies Field Membership Relation
implied universal instantiation ( IUI )
Set Name (Variable)
 A Set is a collection of Mathematical Objects, minimally structured by a
membership Relation, without need of any particular Order.
The Set Extension Representation ( SER ) is adequate to denote the
Set, yet can be rendered more resilient ( less ambiguous ) by added
specification
Domain Quantification
Image Map
Image Map
Implicit Function Free variable equation,
capable of quantification, which can be
evaluated to a truth value
Conjunctive Delineator
Image Set
> Named Set
> Single Free Variable - IUI
( Implicit Universal Instantiation )
> Predicate Ratified Field
Membership ( PRFM ) w/ IUI
> Compound Conjunctive
Predicate
> Implicit Function, Free Variable
Equation Image Map Specification
> Finite Roster Image Set

65. © Art Traynor 2011
Mathematics
Discrete Structures
Usage Conventions for SBN
Set Builder Notation
S = { x | x ℝ  x > 0 } → { ℝ+ }
( Logical ) Predicate
Conditional Separator
Argument
Argument
Can be generalized to constitute
a “ Term ” or “ Expression ”
Predicate (Bifurcated)
Compound expression composed of
a quantification clarifying a Field
Membership Relation with
conjunctive IF-FV image map
specification
Domain Quantification
Ratifies Field Membership Relation
implied universal instantiation ( IUI )
Set Name (Variable)
 A Set is a collection of Mathematical Objects, minimally structured by a
membership Relation, without need of any particular Order.
The Set Extension Representation ( SER ) is adequate to denote the
Set, yet can be rendered more resilient ( less ambiguous ) by added
specification
Domain Quantification
Image Map
Conjunctive Delineator
Image Set
> Named Set
> Single Free Variable - IUI
( Implicit Universal Instantiation )
> Predicate Ratified Field
Membership ( PRFM ) w/ IUI
> Compound Conjunctive
Predicate
> Implicit Function, Free Variable
Inequality Image Map Specification
> Field Image Set
( Lower Bounded )
Image Map
Implicit Function-Free Variable ( IF-FV )
Inequality, capable of quantification,
which can be evaluated to a truth value

66. © Art Traynor 2011
Mathematics
Discrete Structures
Usage Conventions for SBN
Set Builder Notation
S = { ( x , y ) | 0 < y < f(x) } → { ? }
( Logical ) Predicate
Conditional Separator
Argument
Argument
Can be generalized to constitute
a “ Term ” or “ Expression ”
Predicate
Abridged by exclusion of explicit
membership Relation, featuring a
Free Variable Function ( FVF )
image map specification
Set Name (Variable)
 A Set is a collection of Mathematical Objects, minimally structured by a
membership Relation, without need of any particular Order.
Image Map
Image Set
> Named Set
> Two Free Variables ( Ordered
Pair ? ) with Implicit Universal
Instantiation ( IUI )
> Abridged Predicate
> Implicit Function, Conjunctive
Chained Inequality Image Map
Specification
> Infinite Roster Image Set
( Lower Bounded )
Image Map
Conjunctive Explicit Function with free
variable constricted inequality capable of
quantification and truth evaluation
Image Set
Interval Representation an be
generalized to constitute a “
Term ” or “ Expression ”
( x, y ) → f ( x, □ ) → ( f (x), y )
This is a curious image map as the function accepts an ordered pair (domain specification)
but only transforms one element of the ordered pair:
The image set will thus populate with non-zero, non-negative ordered pairs, excluding any
pairs where f (x) ≤ y .

67. © Art Traynor 2011
Mathematics
Discrete Structures
Usage Conventions for SBN
Set Builder Notation
S = { ( k ) | ∃ n ℕ , k = 2n } → { 0, 2, … n }
( Logical ) Predicate
Conditional Separator
Argument
Argument
Can be generalized to constitute
a “ Term ” or “ Expression ”
Predicate
Compound expression composed
of a quantification clarifying a
Fixed Variable Field Membership
Relation with implicit free variable
image map specification
Set Name (Variable)
 A Set is a collection of Mathematical Objects, minimally structured by a
membership Relation, without need of any particular Order.
Image Map
Image Set
> Named Set
> One Free Variable by Implicit
Universal Instantiation ( IUI )
> Compound Predicate,
Comma Delineated
( Implicit Conjunctive )
> Implicit Function, Free Variable
Equation Image Map Specification
> Infinite Roster Image Set
( Lower Bounded )
Image Map
Implicit Function-Free Variable
( IF-FV ) Equation capable of
quantification and truth evaluation
Image Set
Infinite roster with lower Bound
Domain Quantification
Declarative Delineator
> Dependent Variable-Predicate
Ratified Field Membership
( DV-PRFM )
This is a curious image map…why is the dependent (fixed) variable being existentially
instantiated? It seems to make much more sense to instantiate universally (right?).
Also, the IUI instantiation of the fixed variable is awkward, why not make the minimal extra
effort and call-out the function explicitly as in: k = f ( n ) = 2n

68. © Art Traynor 2011
Mathematics
Discrete Structures
Usage Conventions for SBN
Set Builder Notation
S = { ( a ) | ∃ ( p , q )ℕ , ( q ≠ 0 ) ⋀ ( aq = p ) } → { ℚ }
( Logical ) Predicate
Conditional Separator
Argument
Argument
Can be generalized to constitute
a “ Term ” or “ Expression ”
Predicate (Bifurcated)
Compound expression composed
of a quantification clarifying a
Fixed Variable Field Membership
Relation with constricted
conjunctive Implicit Function ( IF )
image map specification
Set Name (Variable)
 A Set is a collection of Mathematical Objects, minimally structured by a
membership Relation, without need of any particular Order.
Image Map
Image Set
> Named Set
> One Free Variable by Implicit
Universal Instantiation ( IUI )
> Compound Predicate,
Comma Delineated
( Implicit Conjunctive )
> Conjunctively constricted Implicit
Function (IF) Equation Image Map
Specification
> Field Image Set
( Unbounded )
Image Map
Conjunctively constricted Implicit
Function ( IF ) Equation capable of
quantification and truth evaluation
Image Set
Unbounded Field
Domain Quantification
Declarative Delineator
> Dependent Variable-Predicate
Ratified Field Membership
( DV-PRFM ) ?
This is another simple but tough SER to penetrate. I had quite a bit of trouble discovering the
identity of the independent (free) variable and its dependent (fixed) adjuncts. In the event, I
relied on an analogy to the previous example where the resultant “ k ” was IUI instantiated
as the independent variable for the equation a = f ( p, q ) = p / q
This establishes this instance as another example of a curious dependent (fixed) variable
argument for an SER…

69. © Art Traynor 2011
Mathematics
Discrete Structures
Cardinality of Sets
Uncountable
Countable
InfiniteFinite
Sets
The key to determining if an infinite set is countable is to establish a one-to-one correspondence
between the set and Z+, the set of positive integers
Either Finite or the same cardinality as Z+
Not Countable
Cardinality |S | = 0
Absolute Value |a | = | – a | establishes a “one-to-one” correspondence with Z+
{Z+} onto even integers, f(n) = 2n establishes a “one-to-one” correspondence with Z+
f(n) = n + 1
good, simple candidate function to establish “one-to-one”
correspondence with Z+

70. © Art Traynor 2011
Mathematics
Discrete Structures
Z+ Mapping Functions
The key to determining if an infinite set is countable is to establish a one-to-one correspondence
between the set and Z+, the set of positive integers
f(n) = |± a | Absolute Value, maps domain set R
to image set Z+ (positive integers)
 establishes a “one-to-one” correspondence
directly with Z+
f(n) = 2n maps domain set R
to image set Ze (even integers)
 ZeZ  R and Z is a countable set (by Ex. 3, pg172) with a
one-to-one correspondence to Z+, and furthermore cardinality
|Z+| =|Z |, and the subset of a countable set is countable
f(n) = n + 1
Art fave: good, simple candidate function to
establish “one-to-one” correspondence with Z+
Useful Mapping Functions for Z+
f(n) = 2n-1 maps domain set R
to image set Zo (odd integers)

ZoZ  R and Z is a countable set (by Ex. 3, pg172) with a
one-to-one correspondence to Z+, and furthermore cardinality
|Z+| =|Z |, and the subset of a countable set is countable
Piecewise defined function
1if ai = 1
1if ai ≠ 1

A string can be denoted by a sequence: a1,a2,… an
and $a piecewise defined function, indexed by Z+
mapping to an arbitrary image set

71. © Art Traynor 2011
Mathematics
Discrete Structures
Definitions
Sets A & B have the same cardinality
if and only if
there is a one-to-one correspondence
from A to B
written |A | = |B |
Cardinality of Sets
#1
injective function will suffice
If there is a one-to-one correspondence
from A to B
the cardinality |A | ≤ |B |
and if the cardinality of A & B are different
then |A | < |B |
#2 injective function will suffice

72. © Art Traynor 2011
Mathematics
Discrete Structures
Definitions
If a set is either finite or
has the same cardinality
as Z+, the set of positive integers,
then that set is countable
Cardinality of Sets
#3
injective function will suffice
A set that is not countable is uncountable
A countably infinite set S
has cardinality |S | = 0

73. © Art Traynor 2011
Mathematics
Discrete Structures
Countability of Sets
Absolute Value |n | = | – n | establishes a “one-to-one” correspondence with Z+
f(n) = 2n ( for even integers ) establishes a “one-to-one” correspondence with Z+
f(n) = n + 1 good, simple candidate function to establish “one-to-one”
correspondence with Z+
Countably Infinite Set s
There are two methods to determine whether an infinite set is countable (both require that a one-to-
one correspondence with Z+, the set of positive integers, be shown)
 Demonstrate that the elements of the set can be sequenced in a list
indexed by Z+, the set of positive integers
 Demonstrate a function mapping from the set to Z+, the set of
positive integers, with at least a one-to-one (injective) correspondence



 Any subset of a countable set is also countable
(see page 173)

74. © Art Traynor 2011
Mathematics
Discrete Structures
Countability of Sets
Countability Techniques
Uncountable
 All elements of R can be represented by a unique decimal expansion
Non-negative
 Each unique decimal expansion of the
individual elements of R can be
sequenced by a matrix R where ri = aij
ri = a0 .a1 a2 a3…
a0 Z+
0 ≤ ai ≤ 9
"r R,ri = Σi = 0
∞ a i
10i
a0 . a11
a1 .
a2 .
am .
.
.
.
a21
a31
am1
.
.
.
a12
a22
a32
am2
.
.
.
a13
a23
a33
am3
.
.
.
. . .
. . .
. . .
. . .
.
.
.
a1n
a2n
a3n
amn
.
.
.
C1 C2 C3 . . . Cn
M = # of Rows
i = Row Number Index
N = # of Columns
j = Column Number Index
nj nj+1 nj+2 nn
mi
mi+1
mi+2
mm
 however $r R where
rx = d0 .d1 d2 d3… and
1 if dii ≠ 1
0 if dii = 1
dx =
thus rx will differ from ri at the jth
position forming an new real number not in R
.
.
.

75. © Art Traynor 2011
Mathematics
Discrete Structures
Countability of Sets
Countability Techniques
Countably Infinite
The elements of Q can be arranged into a sequence,
t = ai , a2 , a3 ,… , an , …
M = # of Rows
i = Row Number Index
N = # of Columns
j = Column Number Index
a1 .
a2 .
am .
.
.
.
. . .
. . .
. . .
. . .
...
C1 C2 C3 . . . Cn
nj nj+1 nj+2 nn
mi
mi+1
mi+2
mm
p1
q1
p2
q1
p3
q1
pn
q1
pi
qi
pi
qi
pi
qi
pi
qi
pi
qi
pi
qi
pi
qi
pi
qi
pi
qi
pi
qi
pi
qi
pi
qi
...
...
...
...
.
.
.

Every rational number can be represented as quotient of two
integers:

pj
qi
pj
qi