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CS355 - Theory of
Computation
Lecture 2: Mathematical
Preliminaries
Set theory
• A set is a collection of objects.
• These objects are referred to as its
elements.
• The order of these elements is not
important.
• The size of a set is its cardinality - | |
• Natural number when the set is finite
•  when the set is infinite
• Empty set has cardinality of zero - |ø| = zero.
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
Membership
• Set membership is denoted using  and non
membership by 


– a
{a , b} and c {a , b}
– apple {apple, pear, banana}
– apple {apples, pears, bananas}


Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
Subsets
• A set A whose elements are also elements of a
set B is called a subset of B
• A B
• A = {0,1,2} and B = {0,2,3,4,1} then A  B
• If |B| > |A| then A is known as a proper subset of
B, denoted by A  B.
• Note, A  A and ø  A for all sets A.
• If A = B then A  B and B  A.
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
Some Elementary Logic
• 
• 
•
• 
•
• 
and
or
implies/only if
not
universal qualifier (for all)
existential qualifier (there exists)
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
More Set Theory
•

union - all the elements of both sets but no
duplicates
•
intersection - all the elements that are in both sets
• - difference (A – B) - the set of elements that are in A
but not in B
•  symmetric difference - the set of elements belonging
to one but not both of two given sets. It is therefore the
union of the complement of A with respect to B and B
with respect to A, and corresponds to the XOR operation
in Boolean logic.

Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
Power Set
• The power set 2n is the set of all possible
subsets of A including ø and A itself.
• If A is finite and consists of n elements,
then the power set has 2n elements.
• e.g. if A = {a,b} then 2A = {ø, {a}, {b}, {a,b}}
• e.g. if A = {a,b,c} then 2A = {ø, {a}, {b}, {c},
{a,b}, {a,c}, {b,c}, {a,b,c}}
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
Relations
• a, b an ordered pair of elements
• It differs from {a,b} in that:
• The order of the elements if important
• The same element may occur twice
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
Cartesian Product
• The Cartesian product (A×B) is the set of all
ordered pairs <x,y> with xA and y B
• Therefore if A = {0,1,2} and B = {c,d} then
A×B = {<0,c>,<0,d>, <1,c>,<1,d>, <2,c>,<2,d>}
B×A = {<c,0>,<c,1>, <c,2>,<d,0>, <d,1>,<d,2>}
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
Binary Relation
• Any subset (call it R) of A×B is called a
binary relation between A and B
– R = {<0 , c>,<1 , d>} is a binary relation
between A and B (A = {0, 1, 2} and B = {c, d})
– Note that the singular ‘binary relation’ relates
to a set of ordered pairs
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
Predicate
• The predicate R(a,b) is true if <a,b>  R
• In future when we talk about a relation R we
mean the combination of the set R and its
implicit predicate R()
• a ^  b where the predicate R(a,b) = true
The domain = {a:a} and
The codomain (range) = {b:b}
• If we have more than one ordered pair
<x1,x2,…,xn> we call it an ordered n-tuple.
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
Functions
• A function f: A → B is a binary relation between
A and B where only one tuple <a,b> exists for
each aA
• f may be written f(a) = b
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
Functions
• Given a function f(a) = b we can say:
–
–
–
–
–
The value of f is a in b
a is the argument and b is the value
f is injective if there is a unique b B for every aA
f is surjective if every b B has an a A
f is bijective if it is both injective and surjective
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
Recap: Number Sequences
• Natural numbers (N): the whole positive numbers
including zero: 0, 1, 2, 3,....
• Integers numbers (Z): the natural numbers plus
their negative counterparts: …,-2,-1,0,1,2,…
• Rational numbers (Q): the integers plus the
rational fractions - those that can be expressed
as the ratio of two integers: ⅓,⅜,…
• Real numbers (R): the rational numbers plus the
irrational numbers: 0.000123, 1.24,…
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
Cardinality and Infinite Sets
• The Cardinality is the number of elements
of a finite set - |A|
• What about infinite sets?
• An infinite set contains infinitely many
elements – can’t write a list of all the
elements of such sets
– Use “…” to mean “continue sequence forever”
– Set of Natural Numbers: {1,2,3,…}
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
Cardinality and Infinite Sets
• Two infinite sets, A and B are
equinumerous/equivalent (A≡B) if a bijection
exists between them.
– Note, we don’t have to be able to compute it in
every instance, just be certain one exists.
• Therefore |A| = |B|  A≡B
• Putting two infinite sets into one-to-one
correspondence is an infinite task, and we
don't pretend that we can do it (that is, finish it)
in finite time.
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
Cardinality and Infinite Sets
• To show that an infinite set, like the even
numbers, can be put into one-to-one
correspondence with another, like the odd
numbers, we need only produce a rule-governed
sequence for each set which runs through the
members without omission or repetition:
– for example, 2, 4, 6... and 1, 3, 5....
• If we can do so, then we know that the nth term
of one sequence will have a counterpart in the
nth term of the other, and vice versa,
guaranteeing one-to-one correspondence for
each element.
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
Cardinality and Infinite Sets
• However there are some infinite sets which are
not equinumerous. We use this to prove that
uncomputable numbers exist.
• A set is countable iff (if and only if) its
cardinality is either finite or equal to N (the
Natural numbers) – i.e. if a bijection exists with
a subset of the natural numbers - A≡N.
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
Cardinality and Infinite Sets
• Other ways to identify a countable set:
– May one order the set such that between any
two elements a, b there is a finite number of
elements?, or
– Could one write out the first two elements of
the set?
• This is possible with N and Z, but not with
R.
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
Cardinality and Infinite Sets
• Finite sets are countable:
– Example: Real numbers with two decimal
places of accuracy between 0 and 1 is a
countable set.
• If a set is not countable it is uncountable.
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
Graphs
• A graph is a set of points with lines connecting
some of the points.
• The points are known as nodes or vertices.
• The connecting lines are known as edges – the
number of edges = degree (each node has
degree 2 and degree 3, respectively below)
1
3
4
5
2
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
Graphs
• A path in a graph is a sequence of nodes
connected by edges.
• A graph is connected if, for every two distinct
nodes a and b, there is a path from a to b.
• A cycle is a path within a graph that starts and
ends at the same node.
• A graph is a tree if it is connected and has no
simple cycles – may contain a specially
designated node called the root.
• Nodes of degree 1 in a tree, apart from the root,
are called the leaves of the tree.
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
Graphs
path
cycle
Leaves
of tree
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
Directed Graph
• If a graph has arrows instead of lines it is a
directed graph.
• The number of arrows leaving a node is
the outdegree of that node and the
number of arrows entering a node is its
indegree.
1
2
3
6
5
4
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
Directed Graph
• In a directed graph an edge from node i to
node j is represented as a pair (i, j)
• The formal description of a directed graph
G is (V, E) where V is the set of nodes and
1
2
E the set of edges.
3
6
5
4
• ({1,2,3,4,5,6}, {(1,2), (1,5), (2,1), (2,4),
(5,4), (5,6), (6,1), (6,3)})
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
Graphs
• In a binary tree each node which is not a
leaf has at most two children.
• If we distinguish between left and right
children then the tree is ordered.
• In a complete tree each node which is not
a leaf has exactly two children.
• A complete tree is perfect if all leaves have
the same height.
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth