Preliminaries
While we assume that the reader has a good working
knowledge of concepts such as sets and functions, we still need to
establish some basic notation and present precise mathematical definitions
for some familiar concepts that are sometimes only treated informally.
The precision will make it easier to make some definitions and prove some
results. However, the reader can decide when, and if, to go over
this material.
Some Typographic conventions
Mathematical symbols are written in blue,
see examples below.
Text that can be copied to, and performed on, UAcalculators is written
in green.
Some basic notation
N denotes
the set of natural numbers {0,1,2,...}
[N] denotes
the set of positive natural numbers {1,2,3,...
}
[n], for
nN,
denotes the set {1,...,n}.
In particular, [0] = ,
the empty set.
Z denotes
the set of integers { ...1, 0, 1,... }
Ordered Pairs
We will have many occasions where we will want to talk about ordered
pairs of elements from a set, X. Given
x,yX,
we will write <x, y> for the ordered
pair with first element x andsecond
element y. The significant
property of ordered pairs is that for any two ordered pairs <x,y>
and
<u,v>
from a set X that we have
<x,y> = <u,v> if, and only if
x=u
and y=v
Note that, we have not said what an ordered pair is, we have only
stated what we require of it. This is enough for most purposes 
as long as you are willing to accept the existence of such things.
However, for the sake of completeness we can define
<x,y> as the set {x, {x,y}}.
This is thedefinition used in many approaches to set theory.
Exercise: Show that defining
<x,y>
as {x,{x,y}} works, i.e., that it satisfies
the above property.
Total and Partial Functions
Definition:
Given sets A and B
a total function, f, from A
to B consists of the following data

The set A, called the source
of f

The set B, called the target
of f

graph(f), a subset of
AB with the property
that for every aA
there exists exactly one bB
such that <a,b>graph(f).
We call graph(f) the graph of f.
We write f:AB
as
an abbreviation for "f is a total function
from A to B".
We write f(a) to denote the unique element
of B such that <a,
f(a)>f
end
Definition:
Given
sets A and B
a partial function, f, from
A
to B consists of the following data

The set A, called the source
of f

The set B, called the target
of f

dom(f), a subset of
A, called the domain of f

fun(f), a total function from dom(f)
to B.
We write f:AB
as
an abreviation for "f is a partial function
from A to B".
We write f(a) to denote the unique element
of B such that <a,
f(a)> is an element of the graph of fun(f)
 note, this implies that adom(f).
end
Definition:
Given total functions
f:AB
and g:BC
we define their composite, written as gf
: AC,
to be the total function from A to C
with graph { <a,c>  there exists
bB such that <a,b>graph(f)
and <b,c> graph(g)}.
Given partial functions f:AB
and g: BC
we define their composite, written as gf
: AC, to be the partial
function from A to C
with

dom(gf)
= { a  adom(fun(f)),
and fun(f)(a)dom(fun(g)) }

graph(fun(.gf))
= { <a,c>  adom(gf)
and fun(g)(fun(f)(a)) = c }
end
Definition:
If f:AB
then we define the image of f , denoted
Im(f), to be { b B
 there exists aA, such that <a,b>graph(f)}
end
Strings
Strings play a basic role in computer science.
However the concept is often treated very informally with a string being
regarded as a sequence of symbols written on a page. While this informal
idea is sufficient for many purposes, we will find it useful, and perhaps
essential, to replace it with a more mathematically precise construct.
Definition:
For any set A,
a string over A
of length n is
a mapping u:[n]A.
Given a string u
over A of length
n we sometimes write it as u = u_{1}u_{2}u_{n}
where u_{i} = u( i ).
Let A* denote the set of all strings
over A.
Let denote the unique
string (mapping) :[0]A,
we call the empty string.
We may sometimes write "" rather than
Given any uA*
let u denote the length of u,
so, u:[u]A.
For any u, vA*
let uv
denote their concatenation, that is the string
uv
:[u+v]A
i
u(i) if iu
i
v(iu) if i>u
end
Definition: Given strings
u:[n]A
and v:[p]A,
a string morphism f:uv
is a mapping f : [n][p]
such that vf
=
u
Sindexed sets, Sary Sets, and Sary Mappings
Given a set S, it is quite common to speak
of Sindexed sets, or sets whose elements
are indexed by S. The usual notation
for an Sindexed set A
would be either A = < a_{s}  sS>
or
A = { a_{s}  sS }.
I prefer the former notation over the latter notation for the simple reason
that an indexed set is not just a set, but a set "with indexing".
While two sets are equal if, and only if, they have the same elements,
there are good reasons for saying that two indexed sets are the same if,
and only if, they have the same indexing. For example, if we
speak of points in an ndimensional space
as being [n]indexed sets of real numbers
(commonly called real vectors) then the order (given by the indexing) is
clearly important since two points are the same if, and only if, they have
the same values at the same indices. On the other hand, the notation
A = { a_{s}  sS }suggests
that A is just a set and that the indexing
need not be considered. Despite all these complications, the notion of
an Sindexed set is commonly used without
bothering to define it. What then is the definition of an Sindexed
set?
Definition:
Given sets S and U
an Sindexed set A
of elements of U is the graph of a mapping
SU
.
end
We shall make considerable use of a specialization of the notion of
an Sindexed set, that we call an Sary
set.
Definition:
Given a set S, and Sary
set is an
Sindexed set of sets (n.b.,
it is not clear what to take for U as there
is no set of all sets).
Given Sary sets A
= < A_{s}  sS>
and B = < B_{s}  sS>
we define an Sary mapping f:AB
to be an Sindexed set of mappings f = < f_{s}:A_{s}B_{s}
 sS>.
end
Commuting Diagrams
A diagram is a directed graph in which the nodes
are labeled with names of sets and whose edges are labeled with names of
functions. A diagram is said to be a commuting diagram if for every
ordered pair of nodes in the diagram, if we compose the functions
along each path from the first node to the second then we always get the
same result. Such a diagram is a convenient way to present various
equalities. For example, we used the diagram
in the definition of string morphism to illustrate
that vf
=
u Sometimes, as in this example, an equals sign,
"=", is inserted in the diagram to indicate that it is a commuting diagram,
but this is not mandatory. Here is another example:
This commuting diagram asserts the commutativity of functional composition,
i.e., that h(gf)
= (hg)f