discrete maths notes.ppt

DISCRETE MATHEMATICS NOTES
ICT1102B/ICT1102D
BIT/DIT YEAR 1, SEMESTER ONE, 2019
BY
KATO DENNIS(KAKEEKA)
&
NAKATO CONSTANCE NAKIMULI(KIRUMBA)

Types of mathematics
Mathematics can be broadly classified into two
categories:
Continuous Mathematics ─ It is based upon continuous
number line or the real numbers. It is characterized by
the fact that between any two numbers, there are
almost always an infinite set of numbers. For example, a
function in continuous mathematics can be plotted in a
smooth curve without breaks.
Discrete Mathematics ─ It involves distinct values; i.e.
between any two points, there are a countable number
of points. For example, if we have a finite set of objects,
the function can be defined as a list of ordered pairs
having these objects, and can be presented as a
complete list of those pairs.
2

Topics in Discrete Mathematics
• Sets, Relations and Functions
• Mathematical Logic & Boolean Algebra
• Group theory
• Counting Theory
• Probability
• Mathematical Induction and Recurrence
Relations
• Graph Theory
• Trees
3

Set Theory
Set theory forms the basis of several other
fields of study like counting theory, relations,
graph theory and finite state machines.
Definition: A set is an unordered collection of
different elements.
5

Representation of a Set
• Sets can be represented in two ways −
– Roster or Tabular Form
– Set Builder Notation
6

Describing a set
1. Roster or Tabular Form/ Explicitly i.e. by listing its
elements using set bracket. The elements are
enclosed within braces and separated by commas.
E.g.
1. A set of vowels in English alphabet,
A = {a,e,i,o,u}
2. A set of odd numbers less than 10,
B = {1,3,5,7,9}
7

Describing set
2. Set Builder Notation
The set is defined by specifying a property that
elements of the set have in common. The set
is described as A = { x : p(x)}
e.g.
1. The set {a,e,i,o,u} is written as:
A = { x : x is a vowel in English alphabet}
2. The set {1,3,5,7,9} is written as:
B = { x : 1≤x<10 and (x%2) ≠ 0}
8

Member of a set
If an element x is a member of any set S, it is
denoted by x∈ S and if an element y is not
a member of set S, it is denoted by y ∉ S.
e.g. If S = {1, 1.2,1.7,2}, 1∈ S but 1.5 ∉S
9

Some Important Sets
N: the set of all natural numbers = {1, 2, 3, 4,
.....}
Z: the set of all integers = {....., -3, -2, -1, 0, 1,
2, 3, .....}
Z+: the set of all positive integers
Q: the set of all rational numbers
R: the set of all real numbers
W: the set of all whole numbers
10

Cardinality of a Set
Cardinality of a set S, denoted by |S|, is the number
of elements of the set.
The number is also referred as the cardinal number.
If a set has an infinite number of elements, its
cardinality is ∞.
e.g. |{1, 4, 3,5}| = 4, |{1, 2, 3,4,5,…}| = ∞
Question: What is the cardinality of the following
sets:
1. the set of all natural numbers
11

Types of Sets
Finite Set: A set which contains a definite number of
elements is called a finite set.
Example: S = {x | x ∈ N and 70 > x > 50}
Infinite Set: A set which contains infinite number of
elements is called an infinite set.
Example: S = {x | x ∈ N and x > 10}
Subset: A set X is a subset of set Y (Written as X ⊆ Y)
if every element of X is an element of set Y.
12

Types of sets
Example subsets
1: Let, X = { 1, 2, 3, 4, 5, 6 } and Y = { 1, 2 }. Here set Y
is a subset of set X as all the elements of set Y is in
set X. Hence, we can write Y ⊆ X.
2: Let, X = {1, 2, 3} and Y = {1, 2, 3}. Here set Y is a
subset (Not a proper subset) of set X as all the
elements of set Y is in set X. Hence, we can write
Y ⊆ X.
13

Types of sets
Proper Subset: The term “proper subset” can be
defined as “subset of but not equal to”. A Set X is a
proper subset of set Y (Written as X ⊂ Y) if every
element of X is an element of set Y and| X| < | Y |.
Example: Let, X = {1, 2, 3, 4, 5, 6} and Y = {1, 2}. Here
set Y ⊂ X since all elements in Y are contained in X
too and X has at least one element is more than set
Y.
14

Types of sets
Universal Set: It is a collection of all elements in a
particular context or application. All the sets in that
context or application are essentially subsets of this
universal set. Universal sets are represented as U.
eg….
Empty Set or Null Set: An empty set contains no
elements. It is denoted by ∅. As the number of
elements in an empty set is finite, empty set is a
finite set. The cardinality of empty set or null set is
zero.
15

Types of sets
e.g. S = {x | x ∈ N and 7 < x < 8} = ∅
Unit Set or Singleton Set: Singleton set or unit set
contains only one element. A singleton set is
denoted by {s}. Example: S = {x | x ∈ N, 7 < x < 9} =
{ 8 }
Equal Set: If two sets contain the same elements
they are said to be equal. Example: If A = {1, 2, 6} and
B = {6, 1, 2}, they are equal as every element of set A
is an element of set B and every element of set B is
an element of set A.
16

Types of sets
Equivalent Set: If the cardinalities of two sets are
same, they are called equivalent sets. E.g: If A = {1, 2,
6} and B = {16, 17, 22}, they are equivalent as
cardinality of A is equal to the cardinality of B. i.e.
|A|=|B|=3
17

Types of sets
Overlapping Set: Two sets that have at least one
common element are called overlapping sets. In case
of overlapping sets:
n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
n(A ∪ B) = n(A - B) + n(B - A) + n(A ∩ B)
n(A) = n(A - B) + n(A ∩ B)
n(B) = n(B - A) + n(A ∩ B)
e.g. Let, A = {1, 2, 6} and B = {6, 12, 42}. There is a
common element ‘6’, hence these sets are
overlapping sets.
18

Types of sets
Disjoint Set: Two sets A and B are called disjoint sets
if they do not have even one element in common.
Therefore, disjoint sets have the following
properties:
n(A ∩ B) = ∅
n(A ∪ B) = n(A) + n(B)
E.g. Let, A = {1, 2, 6} and B = {7, 9, 14}; there is not a
single common element, hence these sets are
Disjoint sets.
19

Venn Diagrams
Venn diagram, invented
in1880 by John Venn, is
a schematic diagram
that shows all possible
logical relations
between different
mathematical sets.
20

Set Operations
Set Operations include :
• Set Union
• Set Intersection
• Set Difference
• Complement of Set
• Cartesian Product.
21

Set Operations
Set Operations include Set Union, Set Intersection, Set
Difference, Complement of Set, and Cartesian Product.
Set Union: The union of sets A
and B (denoted by A ∪ B) is
the set of elements which are
in A, in B, or in both A and B.
Hence,
A∪B = {x | x ∈A OR x ∈B}.
e.g. If A = {10, 11, 12, 13} and
B = {13, 14, 15}, then A ∪ B =
{10, 11, 12, 13, 14, 15}.
Note: The common element
occurs only once 22

Set operations
Set Intersection: The intersection of sets A and B
(denoted by A ∩ B) is the set of elements which are
in both A and B. Hence, A∩B = {x | x ∈A AND x ∈B}.
E.G. If A = {11, 12, 13} and B = {13, 14, 15}, then A∩B
= {13}.
23

Set operations
Set Difference/ Relative Complement: The set
difference of sets A and B (denoted by A–B) is the set
of elements which are only in A but not in B. Hence,
A−B = {x | x ∈A AND x ∉B}. Example: If A = {10, 11,
12, 13} and B = {13, 14, 15}, then (A−B) = {10, 11, 12}
and (B−A) = {14,15}. Here, we can see (A−B) ≠ (B−A)
24

Set operations
Complement of a Set: The complement of a set A
(denoted by A’) is the set of elements which are not
in set A. Hence, A' = {x | x ∉A}.
More specifically, A'= (U–A) where U is a universal set
which contains all objects.
Example: If A ={x | x belongs to set of odd integers}
then A' ={y | y does not belong to set of odd
integers}
25

Set operations
Cartesian Product / Cross Product: The Cartesian
product of n number of sets A1, A2.....An, denoted as
A1 × A2 ×..... × An, can be defined as all possible
ordered pairs (x1,x2,....xn) where x1∈ A1 , x2∈ A2 ,
...... xn ∈ An.
E.G. Take two sets A= {a, b} and B= {1, 2}, The
Cartesian product of A and B is written as: A×B= {(a,
1), (a, 2), (b, 1), (b, 2)}
The Cartesian product of B and A is written as: B×A=
{(1, a), (1, b), (2, a), (2, b)}
26

Power Set
Power set of a set S is the set of all subsets of S including
the empty set. The cardinality of a power set of a set S of
cardinality n is 2n. Power set is denoted as P(S).
27

Power set
E.G. For a set S = {a, b, c, d}
Subsets with 0 elements: {∅} (the empty set)
Subsets with 1 element: {a}, {b}, {c}, {d}
Subsets with 2 elements: {a,b}, {a,c}, {a,d}, {b,c},
{b,d},{c,d}
Subsets with 3 elements: {a,b,c},{a,b,d},{a,c,d},{b,c,d}
Subsets with 4 elements: {a,b,c,d}
Hence, P(S) = { {∅},{a}, {b}, {c}, {d},{a,b}, {a,c}, {a,d},
{b,c}, {b,d},{c,d},{a,b,c},{a,b,d},{a,c,d},{b,c,d},{a,b,c,d} }
| P(S) | = 24 =16.
what is | P ({∅}) | ?
28

Partitioning of a Set
Partition of a set, say S, is a collection of n disjoint
subsets, say P1, P2,...… Pn, that satisfies the following
three conditions: Pi does not contain the empty set.
[ Pi ≠ {∅} for all 0 < i ≤ n]
The union of the subsets must equal the entire original
set. [P1 ∪ P2 ∪ .....∪ Pn = S]
The intersection of any two distinct sets is empty.
[Pa ∩ Pb ={∅}, for a ≠ b where n ≥ a, b ≥ 0 ]
Example: Let S = {a, b, c, d, e, f, g, h} One probable
partitioning is {a}, {b, c, d}, {e, f, g,h}
29

Exercise
1. If A = {1, 2, 6} and B = {16, 17, 22}, with reasons state
whether A and B are equivalent?
b) If U = {1, 2, …, 15}, A = {10, 11, 12, 13} and
B = {13, 14, 15}, find (i)(A−B) or A B’ (ii)A’UB’
2. For a set S = {a, b, c, d }, find (i) the power set P(S)
3. Show that A={1, 2, 5, 7} is not a subset of
B = {x: x , x , x is odd}
4. Represent on a Venn diagram (i) A∩ (B UC)
(ii) (A – B)U(B - A) (iii) (AUBUC)’
5. Given A = {1, 2}, B ={a, b, c} and C ={c, d} ,
D ={3,4} Find (i) (A xB)∩(AxC) (ii) Ax(B∩C)
(iii) AxBxD
30

Unit 2
Relations/ Binary
relations

Definition of relations
A binary relation R from set x to y (written as xRy or
R(x, y)) is a subset of the Cartesian
product x × y. If the ordered pair of G is reversed, the
relation also changes.
A binary relation R on a single set A is a subset of
A × A.
For two distinct sets, A and B, having cardinalities m
and n respectively, the maximum cardinality of a
relation R from A to B is mn.
32

Domain and Range
There are two sets A and B, and relation R have
order pair (x, y), then:
The domain of R, Dom(R), is the set { x | (x, y) ∈
R for some y in B }
The range of R, Ran(R), is the set { y | (x, y) ∈ R
for some x in A }
Let, A = {1,2,9} and B = {1,3,7}
Case 1: If relation R is ‘less than’ then R = {(1,
3), (1, 7), (2, 3), (2, 7)}, Dom(R) = { 1, 2}, Ran(R)
= { 3, 7}
Case 2: If relation R is ‘greater than’ then R =
{(2, 1), (9, 1), (9, 3), (9, 7)} Dom(R) = { 2, 9},
Ran(R) = { 1, 3, 7} 33

Representation of Relations
Using Graph
A relation can be represented using a directed graph.
The number of vertices in the graph is equal to the
number of elements in the set from which the
relation has been defined. For each ordered pair
(x, y) in the relation R, there will be a directed edge
from the vertex ‘x’ to vertex ‘y’. If there is an ordered
pair (x, x), there will be self- loop on vertex ‘x’.
34

Directed graph/ arrow diagram
Suppose, there is a
relation R = {(1, 1), (1,2),
(3, 2)} on set S = {1,2,3}, it
can be represented by the
following directed graph:
Let A = {1, 2, 3} and B = {x, y,
z} and R be the relation from
A to B such that R= {(1, x), (1,
z), (2, y) (3, y)}. The arrow
diagram of the relation is
represented as:
1
2
3
x
y
z
A B
Arrow diagram
Directed graph
1 2
3
35

Representing binary relations
matrix of relation
We can represent a
binary relation R by a
table showing (marking)
the ordered pairs of R.
Example:
Let A = {0, 1, 2},
B = {u,v} and
R = { (0,u), (0,v), (1,v),
(2,u) }
36

Types of Relations
1. The Empty Relation between sets X and Y, or on E,
is the empty set ∅
2. The Full Relation between sets X and Y is the set
X×Y
3. The Identity Relation on set X is the set
{(x,x) | x ∈ X}
4. The Inverse Relation R' of a relation R is defined
as: R’= {(b,a) | (a,b) ∈R}
E.g.: If R = {(1, 2), (2,3)} then R’ will be {(2,1), (3,2)}
37

Types of relations
5. A relation R on set A is called Reflexive if ∀a∈A is
related to a (aRa holds).
e.g. : The relation R = {(a,a), (b,b)} on set X={a,b} is
reflexive
6. A relation R on set A is called Irreflexive if no a∈A
is related to a (aRa does not hold). E.g.: The relation
R = {(a,b), (b,a)} on set X={a,b} is irreflexive
7. A relation R on set A is called Symmetric if xRy
implies yRx, ∀x∈A and ∀y∈A. E.g.: The relation
R = {(1, 2), (2, 1), (3, 2), (2, 3)} on set A={1, 2, 3} is
symmetric.
38

Types of relations
8. A relation R on set A is called Anti-Symmetric if
xRy and yRx implies x=y ∀x ∈ A and ∀y ∈ A. E.g.:
The relation R = { (x,y) ∈ N | x ≤ y } is anti-
symmetric since x ≤ y and y ≤ x implies x = y.
9. A relation R on set A is called Transitive if x Ry
and yRz implies xRz, ∀x,y,z ∈ A. e.g. The relation R
= {(1, 2), (2, 3), (1, 3)} on set A= {1, 2, 3} is
transitive.
10. A relation is an Equivalence Relation if it is
reflexive, symmetric, and transitive
39

Types of relations
Example: The relation R = {(1, 1), (2, 2), (3, 3), (1,
2),(2,1), (2,3), (3,2), (1,3), (3,1)} on set A= {1, 2, 3} is an
equivalence relation since it is reflexive, symmetric, and
transitive.
Partial ordering relations: A relation R on a set S
is called a partial ordering if R is reflexive, anti-
symmetric and transitive.
Note: Read about closure properties of relations
40

Composition of relations
Let A, B and C be sets and R be a relation from
A to B and S be a relation from B to C. Then R
and S give rise to a relation from A to C
denoted by RoS and is defined as
RoS = {(a, c): there exists bϵB for which (a, b)ϵ
R and (b, c)ϵ S}
Question: Let A = {1, 2, 3, 4} , B = {a, b, c, d},
C = {x, y, z} and let R ={(1,a), (2, d), (3, a), (3,
b), (3, d)} and S = {(b, x), (b, z), (c, y), (d, z)}.
Find Ros?
41

Combining relations
Definition: Let A and B be sets. A binary relation
from A to B is a subset of a Cartesian product
A x B.
Let R is a subset of A x B means R is a set of
ordered pairs of the form (a, b) where a ϵ A and
bϵ B.
Relations are sets → combinations via set
operations. Set operations of: union, intersection,
difference and symmetric difference.
42

Combining relations
Example:
1. Let A = {1,2,3} and B = {u, v} and
R1 = {(1,u), (2,u), (2,v), (3,u)}
R2 = {(1,v),(3,u),(3,v)} What is: (i) R1 U R2 = ?
(ii) R1∩R2?
2. Let A = {1,2,3} and B = {u, v} and
R1 = {(1,u), (2,u), (2,v), (3,u)}
R2 = {(1,v),(3,u),(3,v)}
What is: (i) R1UR2 (ii) R1∩R2 (iii) R1 - R2 = ?
43

Exercise 2
1. a) Given sets A= {a, b}, B= {1, 2, 3} and C= {2, 3, 5};
Find (AxB)∩ (AxC)
2. Given A = { 1, 2, 3, 4} and B = { x, y, z}. Let R be the
following relation from A to B: R = { (1, y), (1, z), (2, x), (2,
y), (3, x), (3, z) }
(ii) Draw the arrow diagram of R
(iii) find the inverse relation R-1 of R
(iv) determine the domain and range of R
3. Given the sets A ={a, b, c} and B ={1,2}. find the
number of (i) elements in AxB (ii) relations from A to B
4. a) Prove that the relation R = {(1, 1), (2, 2), (3, 3),
(1, 2),(2,1), (2,3), (3,2), (1,3), (3,1)} on set A= {1, 2, 3} is
reflexive, symmetric, transitive and equivalence
b) Draw directed graph representing R 44

Exercise 2
5. Let A = {1,2,3}, B = {0,1,2} and C = {a,b}.
R = {(1,0), (1,2), (3,1),(3,2)}, S = {(0,b),(1,a),(2,b)}
What is S o R = ?
6. Let A = {1,2,3,4,6} and let R be the relation on
A defined by “x divides y” writen as x|y.
a) write R as a set of ordered pairs
b) Draw its directed graph
c) Find R-1 of R, Describe R-1 in words
45

Introduction of function
A Function assigns to each element of a set,
exactly one element of a related set.
Functions find their application in various
fields like representation of the computational
complexity of algorithms, counting objects,
study of sequences and strings, to name a few.
47

Definition
A function or mapping (Defined as f: X→Y) is a
relationship from elements of one set X to
elements of another set Y (X and Y are non-empty
sets). X is called Domain and Y is called
Codomain of function ‘f’.
Function ‘f’ is a relation on X and Y such that for
each x ∈ X, there exists a unique y ∈ Y
such that (x, y) ∈ R. ‘x’ is called pre-image and ‘y’
is called image of function f.
A function can be one to one or many to one but
not one to many.
48

Injective / One-to-one function
A function f: A→B is injective or one-to-one
function if for every b ∈ B, there exists at most
one a ∈ A such that f(s) = t.
This means a function f is injective if a1 ≠ a2
implies f(a1) ≠ f(a2).
Example
1. f: N →N, f(x) = 5x is injective.
2. f: N→N, f(x) = x2 is injective.
3. f: R→R, f(x) = x2 is not injective as (-x)2 = x2
49

Surjective / Onto function
A function f: A →B is surjective (onto) if the
image of f equals its range. Equivalently, for every
b ∈ B, there exists some a ∈ A such that f(a) = b.
This means that for any y in B, there exists some x
in A such that y = f(x).
Example
1. f : N→N, f(x) = x + 2 is surjective.
2. f : R→R, f(x) = x2 is not surjective since we
cannot find a real number whose square is
negative.
50

Bijective / One-to-one
A function f: A →B is bijective or one-to-one
correspondent if and only if f is both injective and
surjective.
Prove that a function f: R→R defined by f(x) = 2x – 3 is
a bijective function.
Explanation: We have to prove this function is both
injective and surjective.
If f(x1) = f(x2), then 2x1 – 3 = 2x2 – 3 and it implies that
x1 = x2.
Hence, f is injective.
Here, 2x – 3= y
So, x = (y+5)/3 which belongs to R and f(x) = y.
Hence, f is surjective.
Since f is both surjective and injective, we can say f is
bijective. 51

Inverse of a Function
The inverse of a one-to-one corresponding function
f :A → B, is the function g : B → A,
holding the following property:
f(x) = y ↔ g(y) = x
The function f is called invertible, if its inverse function
g exists.
Example:
A function f : Z → Z, f(x) = x + 5, is invertible since it has
the inverse function g :
Z→ Z, g(x) = x – 5
A function f : Z → Z, f(x) = x2 is not invertible since this
is not one-to-one as (-x)2 = x2.
52

Composition of Functions
Two functions f: A→B and g: B→C can be
composed to give a composition g o f. This is a
function from A to C defined by (gof)(x) = g(f(x))
Example: Let f(x) = x + 2 and g(x) = 2x + 1, find
( f o g)(x) and ( g o f)(x)
Solution
(f o g)(x) = f (g(x)) = f(2x + 1) = 2x + 1 + 2 = 2x + 3
(g o f)(x) = g (f(x)) = g(x + 2) = 2 (x+2) + 1 = 2x + 5
Hence, (f o g)(x) ≠ (g o f)(x)
53

Composite functions
Some Facts about Composition
If f and g are one-to-one then the function (g
o f) is also one-to-one.
If f and g are onto then the function (g o f) is
also onto.
Composition always holds associative property
but does not hold commutative property.
54

Exercise 3
1. Let X = {1, 2, 3, 4}. Determine whether or not each relation is a function from X
into X.
f = {(2,3), (1, 4), (2, 1), (3, 2), (4, 4)}.
g = {(3, 1), (4, 2), (1, 1)}.
h = {(2, 1), (3, 4), (1, 4), (2, 1), (4, 4)
2. Determine if each function is one -to-one
a) Te each person on the earth asign the number corresponding to his age.
b) To each book written by only one author assign the author.
c) To each country in the world with president assign its president.
3. Let the function f: A→B, g:B→C, and h:C→D be defined as
f = {(a, 2), (b, 1), (c, 2)}, g ={(1, y), (2, x), (3, w)}, h ={(x, 4), (y, 6), (z, 4), (w, 5)}
a) Draw the arrow diagram to the relations of functions f, g and h.
b) determine if each function is onto
c) find the composition function h o g o f
4. with aid of examples, explain the terms injective, surjective and bijective as
applied to functions.
5. Let f(x) = 2x + 3 and g(x) = 5x - 1 , find (i) ( f o g ) (x) (ii) (g o f)(x)
(iii) g-1(5)
55

Introduction
The rules of mathematical logic specify methods
of reasoning mathematical statements.
Greek philosopher, Aristotle, was the pioneer of
logical reasoning. Logical reasoning provides the
theoretical base for many areas of mathematics
and consequently computer science. It has many
practical applications in computer science like
design of computing machines, artificial
intelligence, definition of data structures for
programming languages etc.
57

Applications of proposition logic
1. Translation of English sentences
2. Inference and reasoning:
– new true propositions are inferred from
existing ones
– Used in Artificial Intelligence:
a) Rule based (expert) systems
b) Automatic theorem provers
3. Design of logic circuit
58

Prepositional Logic – Definition
A proposition is a collection of declarative
statements that has either a truth value "true”
or a non truth value "false". A propositional consists
of propositional variables and connectives.
We denote the propositional variables by capital
letters (A, B, etc). The connectives connect the
propositional variables.
59

Propositions
Some examples of Propositions are given below:
"Man is Mortal", it returns truth value “TRUE”
"12 + 9 = 3 – 2", it returns truth value “FALSE”.
The following is not a Proposition: "A is less than 2".
It is because unless we give a specific value of A, we
cannot say whether the statement is true or false.
Other Not propositions:
• Do you like this class?
• There are x students in this class.
60

Connectives
In propositional logic generally we use five
connectives which are:
Disjunctions/ OR (V)
Conjunctions /AND (Λ)
Negation/ NOT (¬)
Implication / if-then (→)
If and only if (⇔)
61

OR (V) connective
The OR operation of
two propositions A and
B (written as A V B) is
true if at least any of
the propositional
variable A or B is true.
A B AVB
T T T
T F T
F T T
F F F
62

AND (Λ) connective
The AND operation
of two propositions
A and B (written as A
Λ B) is true if both
the propositional
variable A and B is
true.
A B A^B
T T T
T F F
F T F
F F F
63

Negation (¬)connective
The negation of a
proposition A (written
as ¬A) is false when A is
true and is true when A
is false.
A ¬ A
T F
F T
64

Implication / if-then (→) connective
An implication A →B is
the proposition “if A,
then B”. It is false if A is
true and B is false. The
rest cases are true.
A B A→B
T T T
T F F
F T T
F F T
65

If and only if (⇔) connective
A ⇔B is bi-conditional
logical connective which
is true when p and q are
same, i.e. both are false
or both are true.
A B A→B
T T T
T F F
F T F
F F T
66

TAUTOLOGY
A Tautology is
a formula
which is
always true
for every value
of its
propositional
variables.
Example:
Prove [(A → B)
Λ A] →B is a
tautology
A B A→B (A → B) Λ A [(A → B) Λ A] →B
T T T T T
T F F F T
F T T F T
F F T F T
67

Contradictions
A Contradiction
is a formula
which is always
false for every
value of its
propositional
variables.
E.g. : Prove
(A V B) Λ [(¬A) Λ (¬B)]
is a contradiction
A B A V B ¬A ¬B (¬A) Λ (¬B) (A V B) Λ [(¬A) Λ (¬B)]
T T T F F F F
T F T F T F F
F T T T F F F
F F F T T T F
68

Contingency
A Contingency is a
formula which has
both some true and
some false values
for every value of its
propositional
variables.
Example: Prove
(A V B) Λ (¬A) a
contingency
A B A V B ¬A (A V B) Λ (¬A)
T T T F F
T F T F F
F T T T T
F F F T F
69

Propositional Equivalences
Two statements X and Y are logically equivalent if any
of the following two conditions hold:
The truth tables of each statement have the same
truth values.
The bi-conditional statement X ⇔ Y is a tautology.
Example: Prove ¬ (A V B) and [(¬A) Λ (¬B)] are
equivalent
70

Equivalence cont’d
Testing by 1st
method
(Matching truth
table):
Here, we can see
the truth values
of ¬ (A V B) and
[(¬A) Λ (¬B)] are
same, hence the
statements are
equivalent.
A B A V B ¬ (A V B) ¬A ¬B [(¬A) Λ (¬B)]
T T T F F F F
T F T F F T F
F T T F T F F
F F F T T T T
71

Equivalence cont’d
Testing by 2nd
method (Bi-
conditionality)
As [¬ (A V B)] ⇔
[(¬A) Λ (¬B)] is a
tautology, the
statements are
equivalent.
A B ¬ (A V B) [(¬A) Λ (¬B)] [¬ (A V B)] ⇔[(¬A) Λ (¬B)]
T T F F T
T F F F T
F T F F T
F F T T T
72

Exercise 4
1. Construct the truth table for (p → q) Ʌ (¬p ↔ q)
2. show that (p Ʌ q) → p is a tautology
3. prove that (p →q) is equivalent to (¬q → ¬p)
4. show that p V ¬p is a tautology
5. show that pɅ ¬p
6. prove that p ↔ q is equivalent to
[ (p → q ) Ʌ(q →p) ]
7. Show that(i) (p Ʌ q)V r = (p V r) Ʌ (q V r) and
(ii) (pV q)Ʌr = (p Ʌ r) V (qɅr)
8. Show that the propositions = (p Ʌ q) V (~ p V(p^ ~ q)) is a
tautology.
9. Show that the propositions = (p^ ~q) Ʌ (~ p V q) is a
contradiction.
73

Predicate logic
Predicate Logic deals with predicates, which are
propositions containing variables.
Predicate Logic – A predicate is an expression of one
or more variables defined on some specific domain.
A predicate with variables can be made a
proposition by either assigning a value to the
variable or by quantifying the variable.
75

Predicate logic
The following are some examples of predicates:
Let E(x, y) denote "x = y"
Let X(a , b, c) denote "a + b + c = 0"
Let M(x, y) denote "x is married to y"
76

proposition function
A proposition function defined on A is an expression
p(x) which has the property that p(a) is true or false for
each aϵA. The set A is called the domain of p(x) and Tp
of all elements of A for which p(a) is true is called the
truth set of p(x). i.e.
Tp ={x:xϵA, p(x) is true} or Tp = {x: p(x)}
example: Find the truth set of eaach of the proposition
function p(x) defined on the set A of positive integers
a) let p(x) be “x +2 > 7”. Its truth set is {x:xϵN, x+2>7}
={6, 7, 8, ……}
77

b) let p(x) be “x+5<3”. Its truth set = {x:xϵN,
x+5<3} = ф
c) let p(x) be “x+5> 1”. Its truth set = {x: xϵN,
x+5>1,} = {1, 2, 3, 4………} = N. ie x is true for all
elements of N.
78

Quantifiers
The variable of predicates is quantified by
quantifiers. There are two types of quantifier in
predicate logic: Universal Quantifier and Existential
Quantifier.
Universal Quantifier
Universal quantifier states that the statements within
its scope are true for every value of the specific
variable. It is denoted by the symbol ∀.
∀x P(x) is read as for every value of x, P(x) is true.
Example: "Man is mortal" can be transformed into
the propositional form ∀x P(x) where P(x) is the
predicate which denotes x is mortal and the universe
of discourse is all men.
79

Universal quantifier
Q1= If { x: xϵA, p(x)} =A then ∀ x P(x) is true;
otherwise ∀x P(x) is false.
The following are examples of universal
quantifiers:
a) {x: xϵN, x+ 4 > 3} because se its true for all
values of xϵN
b) Let A = {1, 2, 3, 4, 5} then {x: xϵA, x+3<10}
is universal quantifier (∀xϵA) (x +3 <10)
80

Existential Quantifier
Existential quantifier states that the statements
within its scope are true for some values of the
specific variable. It is denoted by the symbol ∃.
∃x P(x) is read as for some values of x, P(x) is true.
Q2= If { x: p(x) ≠ ф then ∃x P(x) is true; otherwise
∃x P(x) is false.
Example:
a) the proposition (∃nϵN)(n +4 < 7) is true since n: n
+4 < 7} = {1, 2} ≠ ф .
b) the proposition (∃nϵN)(n +6 < 4) is false since (n:
n+6 < 4} = ф
81

Nested Quantifiers
If we use a quantifier that appears within the scope
of another quantifier, it is called nested
quantifier.
Examples
∀a ∃b P (x, y) where P (a, b) denotes a + b=0
∀a ∀b ∀c P (a, b, c) where P (a, b) denotes
a + (b+c) = (a+b) +c
Note: ∀a ∃b P (x, y) ≠ ∃a ∀b P (x, y)
82

Exercise 5
1. Let p(x) denote the sentence “x +2 > 5”. State whether or not
p(x) is a propositional function on each of the following sets:
a) N, the set of positive integers; b) M = { -1, -2, -3, …}
2. Determine the truth value of each of the following
statements where U = {1, 2, 3} is the universal set:
a) ∃x ∀y, x2 < y +1; b) ∀x ∃y, x2 + y2 < 12;
c) ∀x ∀ y, x2 +y2 < 12.
3. Let A = {1, 2, …, 9, 10}. Consider each of the following
sentences. If it is a statement, then determine its truth
value. If it is a propositional function, determine its truth set.
a) (∀x ϵ A) (∃y ϵ A)( x +y < 14) b) (∀y ϵ A)( x +y < 14)
c) (∀x ϵ A) (∀y ϵ A)( x +y < 14) d) (∃y ϵ A)( x +y < 14)
83

Boolean algebra
Boolean algebra is algebra of logic. It deals with
variables that can have two discrete values, 0
(False) and 1 (True); and operations that have
logical significance. The earliest method of
manipulating symbolic logic was invented by
George Boole and subsequently came to be
known as Boolean Algebra.
Boolean algebra has now become an
indispensable tool in computer science for its
wide applicability in switching theory, building
basic electronic circuits and design of digital
computers.
85

Boolean function
A Boolean function is a special kind of
mathematical function f: Xn → X of degree n,
where X = {0, 1} is a Boolean domain and n is a
non-negative integer. It describes the way how to
derive Boolean output from Boolean inputs.
Example: Let, F(A, B) = A’B’. This is a function of
degree 2 from the set of ordered pairs of Boolean
variables to the set {0, 1} where F(0, 0) = 1, F(0, 1)
= 0, F(1, 0) = 0 and F(1, 1) = 0
86

Boolean Expressions
A Boolean expression always produces a
Boolean value. A Boolean expression is
composed of a combination of the Boolean
constants (True or False), Boolean variables
and logical connectives. Each Boolean
expression represents a Boolean function.
Example: AB’C is a Boolean expression.
87

Boolean Identities
Double Complement Law
~(~A) = A
Complement Law
A + ~A = 1 (OR Form)
A · ~A = 0 (AND Form)
Idempotent Law
A + A = A (OR Form)
A · A = A (AND Form)
Identity Law
A + 0 = A (OR Form)
A · 1 = A (AND Form)
Dominance Law
A + 1 = 1 (OR Form)
A · 0 = 0 (AND Form)
Commutative Law
A + B = B + A (OR Form)
A· B = B · A (AND Form)
Associative Law
A + (B + C) = (A + B) + C (OR Form)
A· (B · C) = (A · B) · C (AND Form)
Absorption Law
A· (A + B) = A
A + (A · B) = A
Simplification Law
A · (~A + B) = A · B
A + (~A · B) = A + B
Distributive Law
A + (B · C) = (A + B) · (A + C)
A · (B + C) = (A · B) + (A · C)
De-Morgan's Law
~(A · B) = ~A + ~B
~(A+ B) = ~A · ~B 88

Canonical Forms
For a Boolean expression there are two kinds of
canonical forms:
1. The sum of minterms (SOM) form
2. The product of maxterms (POM) form
The Sum of Minterms (SOM) or Sum of Products (SOP)
form
A minterm is a product of all variables taken either in
their direct or complemented form.
Any Boolean function can be expressed as a sum of its
1-minterms and the inverse of the function can be
expressed as a sum of its 0-minterms. Hence, F (list of
variables) = Σ (list of 1-minterm indices)
and F’ (list of variables) = Σ (list of 0-minterm indices)
89

SOP
Example
Let, F(x, y, z) = x’ y’ z’ + x y’ z + x y z’ + x y z
Or, F(x, y, z) = m0 + m5 + m6 + m7
Hence,
F(x, y, z) = Σ (0, 5, 6, 7)
Now we will find the complement of F(x, y, z)
F’ (x, y, z) = x’ y z + x’ y’ z + x’ y z’ + x y’ z’
Or, F’(x, y, z) = m3 + m1 + m2 + m4
Hence, F’(x, y, z) = Σ (3, 1, 2, 4) = Σ (1, 2, 3, 4)
91

POM/POS
The Product of Maxterms (POM) or Product of Sums
(POS) form
A maxterm is addition of all variables taken either in
their direct or complemented form.
Any Boolean function can be expressed as a product of
its 0-maxterms and the inverse of
the function can be expressed as a product of its 1-
maxterms. Hence,
F (list of variables) = π (list of 0-maxterm indices)
and F’(list of variables) = π (list of 1-maxterm indices).
92

POS
Example
Let, F(x, y, z) = (x+y+z) • (x+y+z’) • (x+y’+z) •
(x’+y+z)
Or, F(x, y, z) = M0 • M1 • M2 • M4
Hence,
F (x, y, z) = π (0, 1, 2, 4)
F’'(x, y, z) = (x+y’+z’) • (x’+y+z’) • (x’+y’+z) •
(x’+y’+z’)
Or, F(x, y, z) = M3 • M5 • M6 • M7
Hence, F ' (x, y, z) = π (3, 5, 6, 7)
94

Logic Gates
Boolean functions are implemented by using
logic gates. The following are the logic gates:
95

NOT Gate
A NOT gate inverts a single bit input to a single
bit of output.
96

AND Gate
An AND gate is a logic gate that gives a high
output only if all its inputs are high, otherwise
it gives low output. A dot (.) is used to show
the AND operation.
97

OR Gate
An OR gate is a logic gate that gives high output if
at least one of the inputs is high. A plus (+) is used
to show the OR operation.
98

NAND & NOR gates
A NAND gate is a logic gate that gives a low output only if
all its inputs are high, otherwise it gives high output.
An NOR gate is a logic gate that gives high output if both
the inputs are low, otherwise it gives low output.
Execise: Draw logical circuits for NAND and NOR gates
99

XOR (Exclusive OR) Gate & X-NOR (Exclusive
NOR) Gate
An XOR gate is a logic gate that gives high output if the
inputs are different, otherwise it gives low output.
An EX-NOR gate is a logic gate that gives high output if
the inputs are same, otherwise it
gives low output.
100

Exercise 6
1. What is the output when 011 and 101 are
passed through the AND gate?
2. What is the output when the numbers 1101
and 1001 are passed through the NOR gate
3. Use truth table to prove that
4.Draw a logical diagram of a circuit whose
outputs are:
(i) X = AB+A’BC +ABC’
(ii) X = (A+B+D)(AC+B)
5. Expand F (A, B, C) = ∑m (1,4,5,6,7) in standard
SOP form.
101

Exercise 6
6. Translate the following truth table to an equation:
7. Translate the following equation to a truth table:
A B C f
0 0 0 1
0 0 1 0
0 1 0 0
0 1 1 0
1 0 0 1
1 0 1 1
1 1 0 0
1 1 1 1
f = ABC’ + A’BC’ + ABC + A’BC
102

Introduction to group theory
Group Theory is a branch of mathematics and
abstract algebra that defines an algebraic structure
named as group. Generally, a group comprises of a
set of elements and an operation over any two
elements on that set to form a third element also in
that set.
104

Operators and Postulates
Any set of elements in a mathematical system
may be defined with a set of operators and
a number of postulates.
A binary operator defined on a set of elements is
a rule that assigns to each pair of elements a
unique element from that set. For example, given
the set A={1,2,3,4,5}, we can say ⊗ is a binary
operator for the operation 𝑐 = 𝑎 ⊗ 𝑏, if it
specifies a rule for finding c for the pair of (a,b),
such that a,b,c ∈ A.
105

Postulates
The postulates of a mathematical system form the basic
assumptions from which rules can be deduced.
The postulates are:
Closure: A set is closed with respect to a binary
operator if for every pair of elements in the set, the
operator finds a unique element from that set.
Example: Let A = { 0, 1, 2, 3, 4, 5, …………. }
This set is closed under binary operator into (*), because
for the operation c = a * b, for any a, b ∈ A, the product
c ∈ A.
The set is not closed under binary operator divide (÷),
because, for the operation c = a ÷ b, for any a, b ∈ A, the
product c may not be in the set A. If a = 7, b = 2, then c =
3.5. Here a, b ∈ A but c ∉ A.
106

Postulates
Associative Laws: A binary operator ⊗ on a set A is
associative when it holds the following property:
( 𝑥 ⊗ 𝑦) ⊗ 𝑧 = 𝑥 ⊗ ( 𝑦 ⊗ 𝑧 ), where x, y, z ∈ A
Example: Let A = { 1, 2, 3, 4 }
The operator plus ( + ) is associative because for any
three elements, x, y, z ∈ A, the
property (x + y) + z = x + ( y + z ) holds.
The operator minus ( - ) is not associative since
( x – y ) – z ≠ x – ( y – z )
107

Postulates
Commutative Laws: A binary operator ⊗ on a
set A is commutative when it holds the following
property: 𝑥 ⊗ 𝑦 = 𝑦 ⊗ 𝑥, where x, y ∈ A
Example: Let A = { 1, 2, 3, 4 }
The operator plus ( + ) is commutative because
for any two elements, x,y ∈ A, the property
x + y = y + x holds.
The operator minus ( - ) is not associative since
x – y ≠ y – x
108

Postulates
Distributive Laws: Two binary operators ⊗ and ⊛
on a set A, are distributive over operator ⊛ when
the following property holds:
𝑥 ⊗ ( 𝑦 ⊛ 𝑧 ) = ( 𝑥 ⊗ 𝑦) ⊛ ( 𝑥 ⊗ 𝑧 ) , where
x, y, z ∈ A
Example: Let A = { 1, 2, 3, 4 }
The operators into ( * ) and plus ( + ) are distributive
over operator + because for any
three elements, x,y,z ∈ A, the property
x * ( y + z ) = ( x * y ) + ( x * z ) holds.
However, these operators are not distributive over *
since
x + ( y * z ) ≠ ( x + y ) * ( x + z )
109

Positulates
Identity Element: A set A has an identity element
with respect to a binary operation ⊗ on A, if there
existsan element 𝑒 ∈ A, such that the following
property holds:
𝑒 ⊗ 𝑥 = 𝑥 ⊗ 𝑒, where x ∈ A
Example: Let Z = { 0, 1, 2, 3, 4, 5, ……………….. }
The element 1 is an identity element with respect to
operation * since for any element x ∈ Z,
1 * x = x * 1
On the other hand, there is no identity element for
the operation minus ( - )
110

Postulates
Inverse: If a set A has an identity element 𝑒 with
respect to a binary operator ⊗, it is said to have
an inverse whenever for every element x ∈ A, there
exists another element y ∈ A, such that the following
property holds: 𝑥 ⊗ 𝑦 = 𝑒
Example: Let A = { ………….. -4, -3, -2, -1, 0, 1, 2, 3, 4,
5, ………….. }
Given the operation plus ( + ) and 𝑒 = 0, the inverse
of any element x is (-x) since x + (- x) = 0
111

Postulates
De Morgan’s Law: De Morgan’s Laws gives a pair of
transformations between union and intersection of
two (or more) sets in terms of their complements.
The laws are: a) (A⋃B)′ = A′⋂ B′
b) (A⋂B)′ = A′⋃ B′
Example: Let A = { 1, 2, 3, 4}, B = {1, 3, 5, 7}, and
Universal set U = { 1, 2, 3, ………, 9, 10 }
A′ = { 5, 6, 7, 8, 9, 10}, B′ = { 2, 4,6,8,9,10}
A ⋃ B = {1, 2, 3,4, 5, 7}, A⋂B = { 1,3},
(A ⋃ B)′ = { 6, 8,9,10}, A′⋂B′ = { 6, 8,9,10}
Thus, we see that (A⋃B)′ = A′⋂ B′
(A ∩ B)′ = { 2,4, 5,6,7,8,9,10}, A′ ∪ B′ = { 2,4,
5,6,7,8,9,10}
Thus, we see that (A⋂B)′ = A′⋃ B′ 112

Semi-group
A finite or infinite set ‘S’ with a binary operation ‘0’
(Composition) is called semi-group if it holds following
two conditions simultaneously:
Closure: For every pair (a, b) ∈ S, (a 0 b) has to be
present in the set S.
Associative: For every element a, b, c ∈ S, (a 0 b) 0 c = a 0
(b 0 c) must hold.
Example: The set of positive integers (excluding zero)
with addition operation is a semi-group. For example,
S = {1, 2, 3,...}
Here closure property holds as for every pair (a, b) ∈ S,
(a + b) is present in the set S. For example, 1 +2 =3 ∈ S]
Associative property also holds for every element a, b, c
∈S, (a + b) + c = a + (b + c).
For example, (1 +2) +3=1+ (2+3)=5 113

Monoid
A monoid is a semigroup with an identity element.
The identity element (denoted by e or E) of a set S is
an element such that (a 0 e) = a, for every element a
∈ S. An identity element is also called a unit element.
So, a monoid holds three properties simultaneously:
Closure, Associative, Identity element.
114

Monoid
Example on monoid
The set of positive integers (excluding zero) with
multiplication operation is a monoid. S = {1, 2, 3,...}
(a × b) is present in the set S.
[For example, 1 × 2 = 2 ∈ S and so on]
∈S, (a × b) × c = a × (b × c)
[For example, (1 × 2) × 3 = 1 × (2 × 3) = 6 and so on]
Identity property also holds for every element a ∈ S, (a ×
e) = a [For example, (2 ×1) = 2, (3 ×1) =3 and so on].
Here identity element is 1.
115

Group
A group is a monoid with an inverse element. The
inverse element (denoted by I) of a set S is an
element such that (a 0 I) = (I 0 a) =a, for each
element a ∈ S. So, a group holds four properties
simultaneously - i) Closure, ii) Associative, iii) Identity
element, iv) Inverse element. The order of a group G
is the number of elements in G and the order of an
element in a group is the least positive integer n such
that an is the identity element of that group G.
116

Group
Examples: The set of N×N non-singular matrices
form a group under matrix multiplication operation.
The product of two N×N non-singular matrices is also
an N×N non-singular matrix which holds closure
property.
Matrix multiplication itself is associative. Hence,
associative property holds.
The set of N×N non-singular matrices contains the
identity matrix holding the identity element
property.
As all the matrices are non-singular they all have
inverse elements which are also nonsingular
matrices. Hence, inverse property also holds. 117

Abelian Group
An abelian group G is a group for which the element
pair (a,b) ∈ G always holds
commutative law. So, a group holds five properties
simultaneously - i) Closure, ii) Associative, iii) Identity
element, iv) Inverse element, v) Commutative.
118

Abelian Group
Example: The set of positive integers (including zero)
with addition operation is an abelian group.
G = {0, 1, 2, 3,…}
(a + b) is present in the set S. [For example, 1 +2 =2 ∈ S
and so on]
∈S, (a + b) + c = a + (b + c)
[For example, (1 +2) +3=1 + (2 +3) =6 and so on]
Identity property also holds for every element a ∈ S, (a ×
e) = a [For example, (2 ×1) =2, (3 ×1) =3 and so on].
Here, identity element is 1.
Commutative property also holds for every element a
∈S, (a × b) = (b × a) [For example,
(2 ×3) = (3 ×2) =3 and so on] 119

Cyclic Group and Subgroup
A cyclic group is a group that can be generated by a
single element. Every element of a cyclic group is a
power of some specific element which is called a
generator. A cyclic group can be generated by a
generator ‘g’, such that every other element of the
group can be written as a power of the generator ‘g’.
120

Cyclic Group and Subgroup
Note: A cyclic group is always an abelian group but
not every abelian group is a cyclic group. The rational
numbers under addition is not cyclic but is abelian.
A subgroup H is a subset of a group G (denoted by
H ≤ G) if it satisfies the four properties
simultaneously: Closure, Associative, Identity
element, and Inverse.
A subgroup H of a group G that does not include the
whole group G is called a proper subgroup (Denoted
by H<G). A subgroup of a cyclic group is cyclic and a
abelian subgroup is also abelian.
121

Partially Ordered Set (POSET)
A partially ordered set consists of a set with a binary
relation which is reflexive, antisymmetric and
transitive. "Partially ordered set" is abbreviated as
POSET.
122

Poset
Examples
1. The set of real numbers under binary operation
less than or equal to (≤) is a poset.
Let the set S = {1, 2, 3} and the operation is ≤
The relations will be {(1, 1), (2, 2), (3, 3), (1, 2), (1, 3),
(2, 3)}
This relation R is reflexive as {(1, 1), (2, 2), (3, 3)} ∈ R
This relation R is anti-symmetric, as
{(1, 2), (1, 3), (2, 3)} ∈ R and {(2,1), (3,1), (3,2)} ∉ R
This relation R is also transitive as {(1,2), (2,3), (1,3)}
∈ R. Hence, it is a poset.
123

Exercise 7
1. Consider the set Q of rational numbers, and let * be the operation on
Q defined by
a* b = a + b – ab
a) Find 3*4, 2*(-5) and 7*1/2
b) Is (Q, *) a semigroup? Is it commutative?
c) Find the identity element for *
d) Do any of the element in Q have inverse? What is it?
2. a) state whether or not each of the following six subsets of the
positive intergers N is closed under the operation of multiplication
(i) A = {0, 1} (ii) B = {1, 2} (iii) C = {x: x is prime}
(iv) D = {2, 4, 6, …} = {x: x is even} (v) E = {1, 3, 5,…} = {x: x is odd} (vi)
F = { 2, 4, 8, …} = {x: x = 2n, nϵN}
b) Which of the six sets in a) above, if any, are closed under the
operation of addition?
3. Given the group G = {1. 2, 3, 4, 5, 6} under multiplication modulo 7.
a) Find the multiplication table G. b) find 2-1, 3-1, 6-1. c) find the
orders and subgroups generated by 2 and 3. d) Is G cyclic?
124

Introduction
In daily lives, many a times one needs to find out the
number of all possible outcomes for a series of
events. For instance, in how many ways can a panel
of judges comprising of 6 men and 4 women be
chosen from among 50 men and 38 women? For
solving these problems, mathematical theory of
counting are used. Counting mainly encompasses
fundamental counting rule, the permutation rule,
and the combination rule.
126

Permutations
A permutation is an arrangement of some elements
in which order matters. In other words a Permutation
is an ordered Combination of elements.
Examples
From a set S ={x, y, z} by taking two at a time, all
permutations are: xy, yx, xz, zx, yz, zy.
We have to form a permutation of three digit
numbers from a set of numbers S= {1, 2, 3}. Different
three digit numbers will be formed when we arrange
the digits. The permutation will be = 123, 132, 213,
231, 312, 32
127

Permutations
The number of permutations of ‘n’ different things
taken ‘r’ at a time is denoted by nPr
𝑛𝑃𝑟 = 𝑛! /(𝑛 − 𝑟)!, where 𝑛! = 1.2.3. … . . (𝑛 − 1). 𝑛
Some important formulas of permutation
1. If there are n elements of which a1 are alike of some
kind, a2 are alike of another kind; a3 are alike of third
kind and so on and ar are of rth kind, where (a1 + a2 + ...
ar) = n. Then, number of permutations of these n objects
is = n! / [ (a1!) (a2!)..... (ar!)].
2. Number of permutations of n distinct elements taking
n elements at a time = nPn = n!
128

Important formulas of permutation
3. The number of permutations of n dissimilar
elements taking r elements at a time, when x
particular things always occupy definite places
= n-xPr-x
elements when r specified things always come
together is: r! (n−r+1)!
elements when r specified things never come
together is: n!–[r! (n−r+1)!]
6. The number of circular permutations of n
different elements taken x elements at time = nPx /x
7. The number of circular permutations of n
different things = nPn /n 129

Permutations
Problem 1: From a bunch of 6 different cards, how
many ways we can permute it?
Solution: As we are taking 6 cards at a time from a
deck of 6 cards, the permutation will be
6P6 = 6! = 720
Problem 2: In how many ways can the letters of the
word 'READER' be arranged?
Solution: There are 6 letters word (2 E, 1 A, 1D and
2R.) in the word 'READER'.
The permutation will be = 6! / [(2!) (1!)(1!)(2!)] =
180.
130

Combinations
A combination is selection of some given
elements in which order does not matter.
The number of all combinations of n things, taken
r at a time is:
𝑛𝐶𝑟 = 𝑛! /𝑟! (𝑛 − 𝑟)!
Problem 1
Find the number of subsets of the set {1, 2, 3, 4,
5, 6} having 3 elements.
Solution
The cardinality of the set is 6 and we have to
choose 3 elements from the set. Here, the
ordering does not matter. Hence, the number of
subsets will be 6C3=20.
131

Combinations
Problem 2
There are 6 men and 5 women in a room. In how
many ways we can choose 3 men and 2 women from
the room?
Solution
The number of ways to choose 3 men from 6 men is
6C3 and the number of ways to choose 2 women from
5 women is 5C2
Hence, the total number of ways is:
6C3 ×5C2=20×10=200
132

Combinations
Problem 3
How many ways can you choose 3 distinct groups of 3
students from total 9 students?
Solution
Let us number the groups as 1, 2 and 3
For choosing 3 students for 1st group, the number of
ways: 9C3
The number of ways for choosing 3 students for 2nd
group after choosing 1st group: 6C3
The number of ways for choosing 3 students for 3rd
group after choosing 1st and 2nd group: 3C3
Hence, the total number of ways = 9C3 ×6C3 × 3C3 =
84×20×1 =1680 133

Pigeonhole Principle
Pigeonhole Principle states that if there are fewer
pigeon holes than total number of pigeons and each
pigeon is put in a pigeon hole, then there must be at
least one pigeon hole with more than one pigeon. If
n pigeons are put into m pigeonholes where n>m,
there's a hole with more than one pigeon.
134

Pigeonhole Principle
Examples
1. Ten men are in a room and they are taking part in
handshakes. If each person
shakes hands at least once and no man shakes the
same man’s hand more than once then two men
took part in the same number of handshakes.
2. There must be at least two people in a class of 30
whose names start with the same alphabet.
135

The Inclusion-Exclusion principle
The Inclusion-exclusion principle computes the
cardinal number of the union of multiple non-disjoint
sets. For two sets A and B, the principle states:
|A ∪B| = |A| + |B| – |A∩B| For three sets A, B and
C, the principle states:
|A∪B∪C | = |A| + |B| + |C| – |A∩B| – A∩C| –
|B∩C| + |A∩B∩C |
The generalized formula:
|⋃ 𝐴𝑖 𝑛𝑖=1 | = Σ |𝐴𝑖 ∩ 𝐴𝑗 | 1≤𝑖<𝑗≤𝑛 + Σ |𝐴𝑖 ∩ 𝐴𝑗 ∩ 𝐴𝑘 | −
… … + (−1)𝑛−1|𝐴1 ∩ … ∩ 𝐴2|1≤𝑖<𝑗<𝑘≤𝑛
136

Inclusion-Exclusion
Problem 1
How many integers from 1 to 50 are multiples of 2 or
3 but not both?
Solution
From 1 to 100, there are 50/2=25 numbers which are
multiples of 2.
There are 50/3=16 numbers which are multiples of 3.
There are 50/6=8 numbers which are multiples of
both 2 and 3.
So, |A|=25, |B|=16 and |A∩B|= 8.
|A ∪ B| = |A| + |B| – |A∩B| =25 + 16 – 8 = 33
137

Inclusion-Exclusion
Problem 2
In a group of 50 students 24 like cold drinks and 36
like hot drinks and each student likes
at least one of the two drinks. How many like both
coffee and tea?
Solution
Let X be the set of students who like cold drinks and
Y be the set of people who like hot drinks.
So, | X ∪ Y | = 50, |X| = 24, |Y| = 36
|X∩Y| = |X| + |Y| – |X∪Y| = 24 + 36 – 50 = 60 – 50 =
10
Hence, there are 10 students who like both tea and
coffee.
138

Exercise 8
1. (a) In how many ways can 11 students be selected from a group of
40 students to stand in line for a picture?
(b) In how many ways can 11 students be selected from a group of
40 students for a school football team?
2. Consider the word IMPLEMENTATION. In how many ways, one can
rearrange the letters of this word?
3. In how many ways can 8 people sit at a round table?
4. In how many ways can 13 cards be selected from a pack of 52
playing cards?
5. A hockey team containing 5 men and 6 women is to be chosen 7
men and 9 women. In how many ways can this be done
6. There 20 books on a shelf, but the red covers of two of them clash,
and they must not be put together. In how many ways can the
books be arranged?
139

Introduction
Probability can be conceptualized as finding the
chance of occurrence of an event.
Mathematically, it is the study of random processes
and their outcomes. The laws of
probability have a wide applicability in a variety of
fields like genetics, weather forecasting, opinion
polls, stock markets etc.
141

Probability
Basic concepts
Random Experiment: An experiment in which all
possible outcomes are known and the exact output
cannot be predicted in advance is called a random
experiment. Tossing a fair coin is an example of
random experiment.
Sample Space: When we perform an experiment,
then the set S of all possible outcomes is called the
sample space. If we toss a coin, the sample space
S = {H, T}
Event: Any subset of a sample space is called an
event. After tossing a coin, getting Head on the top is
an event.
142

Basic concepts of probability
The word "probability" means the chance of
occurrence of a particular event. The best we
can say is how likely they are to happen, using the
idea of probability.
Probability of occurrence of an event =
Total number of favourable outcome/Total number
of Outcomes
As the occurrence of any event varies between 0%
and 100%, the probability varies between 0 and 1.
143

Probability
Steps to find the probability:
Step 1: Calculate all possible outcomes of the
experiment.
Step 2: Calculate the number of favorable outcomes
of the experiment.
Step 3: Apply the corresponding probability formula.
144

Probability
Tossing a Coin
If a coin is tossed, there are two possible outcomes:
Heads (H) or Tails (T).
So, Total number of outcomes = 2
Hence, the probability of getting a Head (H) on top is
½ and the probability of getting a Tails (T) on top is ½
145

Probability
Throwing a Dice
When a dice is thrown, six possible outcomes can be
on the top: 1, 2, 3, 4, 5, 6.
The probability of any one of the numbers is 1/6.
The probability of getting even numbers is 3/6=1/3.
The probability of getting odd numbers is 3/6=1/3.
146

Probability
Taking Cards From a Deck
From a deck of 52 cards, if one card is picked find the
probability of an ace being drawn and also find the
probability of a diamond being drawn.
Total number of possible outcomes: 52
Outcomes of being an ace: 4
Probability of being an ace = 4/52 =1/13
Probability of being a diamond = 13/52 =1/4
147

Probability Axioms
The probability of an event always varies from 0 to 1.
[0 ≤ P(x) ≤ 1]
For an impossible event the probability is 0 and for a
certain event the probability is 1.
If the occurrence of one event is not influenced by
another event, they are called mutually exclusive or
disjoint.
If A1, A2....An are mutually exclusive/disjoint events,
then P(Ai ∩ Aj) = ϕ for i≠j and P(A1 ∪ A2 ∪.... An) =
P(A1) + P(A2)+..... P(An)
148

Properties of Probability
1. If there are two events x and x̅ which are
complementary, then the probability of the
complementary event is: P(x̅) = 1– P(x)
2. For two non-disjoint events A and B, the
probability of the union of two events:
P(A∪ B) = P(A) + P(B)
3. If an event A is a subset of another event B
(i.e. A ⊂ B), then the probability of A is less than or
equal to the probability of B. Hence, A ⊂ B implies
P(A) ≤ p(B)
149

Conditional Probability
The conditional probability of an event B is the
probability that the event will occur given
an event A has already occurred. This is written as
P(B|A).
Mathematically: P(B|A) = P(A ∩ B) / P(A)
If event A and B are mutually exclusive, then the
conditional probability of event B after
the event A will be the probability of event B that is
P(B).
150

Conditional probability
Problem 1
In a country 50% of all teenagers own a cycle and
30% of all teenagers own a bike and cycle. What is
the probability that a teenager owns bike given that
the teenager owns a cycle?
Solution
Let us assume A is the event of teenagers owning
only a cycle and B is the event of teenagers owning
only a bike. So, P(A) = 50/100 = 0.5 and P(A ∩ B) =
30/100= 0.3 from the given problem. P(B|A) =
P(A ∩ B) / P(A) = 0.3/0.5 = 0.6
Hence, the probability that a teenager owns bike
given that the teenager owns a cycle is 60%.
151

Problem 2
In a class, 50% of all students play cricket and 25% of
all students play cricket and volleyball. What is the
probability that a student plays volleyball given that
the student plays cricket?
Solution
Let us assume A is the event of students playing only
cricket and B is the event of students playing only
volleyball.
So, P(A) = 50/100=0.5 and P(A ∩ B) = 25/100=0.25
from the given problem.
152

P(B|A) = P(A ∩ B) / P(A) =0.25/0.5 =0.5
Hence, the probability that a student plays volleyball
given that the student plays cricketis 50%.
Problem 3: Six good laptops and three defective
laptops are mixed up. To find the defective laptops all
of them are tested one-by-one at random. What is
the probability to find both of the defective laptops
in the first two pick?(answer 1/21)
153

Bayes' Theorem
Theorem: If A and B are two mutually exclusive
events, where P(A) is the probability of A and P(B) is
the probability of B, P(A | B) is the probability of A
given that B is true. P(B | A) is the probability of B
given that A is true, then Bayes’ Theorem states:
P(A | B) = P(B | A) P(A) Σ P(B | Ai)P(Ai)
In situations where all the events of sample space
are mutually exclusive events.
In situations where either P( Ai ∩ B ) for each Ai or
P( Ai ) and P(B|Ai ) for each Ai is known
154

Application of Bayes’ Theorem
Problem
Consider three pen-stands. The first pen-stand
contains 2 red pens and 3 blue pens; the second one
has 3 red pens and 2 blue pens; and the third one
has 4 red pens and 1 blue pen. There is equal
probability of each pen-stand to be selected. If one
pen is drawn at random, what is the probability that
it is a red pen?
155

Bayes’ Theorem
According to Bayes’ Theorem,
P(B) = P(A1).P(B|A1) + P(A2).P(B|A2) + P(A3).P(B|A3)
= 1/3 ∙ 2/5 + 1/3 ∙ 3/5 + 1/3 ∙ 4/5 = 3/5
NOTE: Read about Random variables, expectations
156

Exercise 9
1. Two unbiased coins are tossed together. What is the
probability (i) that both are tails (ii) at least a tail is
ontop
2. Two dice are tossed together. What is the
probability that (i) at least a prime number appears
on top (ii) the sum of the two numbers on top is
odd (ii) the sum of the two numbers is greater than
8.
3. A die and a coin are tossed together. What is the
probability that a prime number and a tail appears
on top.
4. Assume the probability of getting a flu is 0.2.
Assume the probability of having a high fever given
the flu: 0.9. What is the probability of getting a flu
with fever? 157

Mathematical Induction
Mathematical induction, is a technique for proving
results or establishing statements for natural
numbers.
Mathematical Induction is a mathematical technique
which is used to prove a statement, a formula or a
theorem is true for every natural number.
The technique involves two steps to prove a
statement, as stated below:
Step 1(Base step): It proves that a statement is true
for the initial value.
Step 2(Inductive step): It proves that if the statement
is true for the nth iteration (or number n), then it is
also true for (n+1)th iteration ( or number n+1). 159

How to Do It
Step 1: Consider an initial value for which the
statement is true. It is to be shown that the
statement is true for n=initial value.
Step 2: Assume the statement is true for any value of
n=k. Then prove the statement is
true for n=k+1. We actually break n=k+1 into two
parts, one part is n=k (which is already
proved) and try to prove the other part.
160

Prove by Induction
Problem 1
3n - 1 is a multiple of 2 for n = 1, 2, ...
Solution
Step 1: For n = 1, 31 - 1 = 3 - 1 = 2 which is a multiple
of 2
Step 2: Let us assume 3n - 1 is true for n=k, Hence,
3k -1 is true (It is an assumption)
We have to prove that 3k+1 - 1 is also a multiple of 2
3k+1 – 1 = 3 × 3k – 1 = (2 × 3k) + (3k –1)
The first part (2×3k) is certain to be a multiple of 2
and the second part (3k -1) is also true as our
previous assumption.
Hence, 3k+1 – 1 is a multiple of 2.
So, it is proved that 3n – 1 is a multiple of 2. 161

Maths induction
Problem 2: 1 + 3 + 5 + ... + (2n-1) = n2 for n=1, 2, ...
Solution
Step 1: For n=1, 1 = 12, Hence, step 1 is satisfied.
Step 2: Let us assume the statement is true for n=k.
Hence, 1 + 3 + 5 + ... + (2k-1) = k2 is true (It is an
assumption)
We have to prove that 1 + 3 + 5 + ... + (2(k+1)-1) = (k+1)2
also holds 1 + 3 + 5 + ... + (2(k+1) – 1)
= 1 + 3 + 5 + ... + (2k+2 – 1) = 1 + 3 + 5 + ... + (2k + 1)
= 1 + 3 + 5 + ... + (2k – 1) + (2k + 1) = k2 + (2k + 1)
= (k + 1)2
So, 1 + 3 + 5 + ... + (2(k+1) – 1) = (k+1)2 hold which
satisfies the step 2.
Hence, 1 + 3 + 5 + ... + (2n – 1) = n2 is proved.
162

Exercise 10
1. Prove that (ab)n = anbn is true for every natural number n.
2. Prove by induction that
1∙2+2∙3+3∙4+⋯+𝑛(𝑛+1) = 𝑛(𝑛+1)(𝑛+2)/3.
3. Using the mathematical induction, verify that the
following statement 1∙(1!)+2∙(2!)+⋯+𝑛∙(𝑛!)=(𝑛+1)!−1
is true for all integers 𝑛≥1
4. Prove by principle of mathematical induction that 3𝑛<𝑛!
for 𝑛>6.
5. Prove that 1+2+3+……+ n = ½n(n + 1).
6. Prove that 12 +22 + 32 + …+ n2 = 1/6n(n+1)(2n+1)
7. Prove by mathematical induction that 3 j (n3 􀀀 n) for
every positive integer n.
163

A recurrence relation
A recurrence relation is an equation that recursively
defines a sequence where the next term is a function
of the previous terms (Expressing Fn as some
combination of Fi with i<n).
Examples:
Fibonacci series: Fn = Fn-1 + Fn-2,
Tower of Hanoi: Fn = 2Fn-1 + 1.
165

Linear Recurrence Relations
A linear recurrence equation of degree k or order k is
a recurrence equation which is in the format
xn= A1 xn-1+ A2 xn-1+ A3 xn-1+... Ak xn-k (An is a constant
and Ak≠0) on a sequence of numbers as a first-
degree polynomial.
166

Solving linear recurrence relation
Suppose, a two ordered linear recurrence relation is:
Fn = AFn-1 +BFn-2 where A and B are real numbers. The
characteristic equation for the above recurrence
relation is: x2 − Ax − B = 0
Three cases may occur while finding the roots:
Case 1: If this equation factors as (x- x1)(x- x2) = 0 and it
produces two distinct real roots x1 and x2, then
Fn = ax1
n+ bx2
n is the solution. [Here, a and b are
constants]
Case 2: If this equation factors as (x- x1)2 = 0 and it
produces single real root x1, then Fn = a x1
n+ bn x1
n is the
solution.
Case 3: If the equation produces two distinct complex
roots, x1 and x2 in polar form x1 = r ∠ θ and x2 = r ∠(- θ),
then Fn = rn (a cos(nθ)+ b sin(nθ)) is the solution. xxx
167

Linear recurrence relation
Problem 1
Solve the recurrence relation Fn = 5Fn-1 - 6Fn-2 where
F0 = 1 and F1 = 4
Solution
The characteristic equation of the recurrence
relation is:
x2 – 5x + 6=0,
So, (x-3) (x-2) = 0
Hence, the roots are:
x1 = 3 and x2= 2
168

linear recurrence relation
The roots are real and distinct. So, this is in the form
of case 1
Hence, the solution is:
Fn = ax1
n+ bx2
n
Here, Fn = a3n+ b2n (As x1 = 3 and x2= 2)
Therefore,
1=F0 = a30+ b20 = a+b
4=F1 = a31+ b21 = 3a+2b
Solving these two equations, we get a = 2 and b = -1
Hence, the final solution is:
Fn = 2.3n + (-1) . 2n= 2.3n - 2n
169

Problem 2
Solve the recurrence relation Fn = 10Fn-1 - 25Fn-2
where F0 = 3 and F1 = 17
Solution
The characteristic equation of the recurrence
relation is: x2 –10x -25 =0,
So, (x – 5)2 = 0
Hence, there is single real root x1 = 5
As there is single real valued root, this is in the form
of case 2
170

Hence, the solution is:
Fn = ax1
n + bnx1
n
3 = F0= a.50+ b.0.50 = a
17 = F1= a.51 + b.1.51 = 5a+5b
Solving these two equations, we get a = 3 and
b = 2/5.
Hence, the final solution is:
Fn = 3.5n + (2/5) .n.2n
171

Exercise 11
1. Solve the recurrence relation
2𝑎𝑛=7𝑎𝑛−1−3𝑎𝑛−2 for the initial conditions 𝑎0=1, 𝑎1=1.
2. Find the solution of the linear homogeneous recurrence
relation of second-degree with constant coefficients
𝑎𝑛=7𝑎𝑛−1−10𝑎𝑛−2 , 𝑎0=2, 𝑎1=3.
3. Given the recurrence relation
𝑎𝑛=−2𝑎𝑛−1+15𝑎𝑛−2 with the initial conditions 𝑎0=1 and
𝑎1=7.
(a)Write the characteristic equation.
(b)Solve the recurrence relation.
4. Find the solution of second-order linear homogeneous
recurrence relation 𝑎𝑛=2𝑎𝑛−1+15𝑎𝑛−2 with 𝑎0=1 and 𝑎1=−4.
172

Graph theory
A graph is a set of points, called nodes or vertices,
which are interconnected by a set of lines called
edges.
The study of graphs, or graph theory is an important
part of a number of disciplines in the fields of
mathematics, engineering and computer science.
174

Graph theory
What is a Graph?
Definition: A graph (denoted as G = (V, E)) consists of
a non-empty set of vertices or nodes V and a set of
edges E.
175

Graphs
A graph with four vertices and four edges
Example: Let us consider, a Graph is G = (V, E)
where V = {a, b, c, d} and E = {{a, b}, {a, c}, {b, c},{c,
d}}
176

Graphs – basic concepts
Degree of a Vertex: The
degree of a vertex V of a
graph G (denoted by
deg (V)) is the number of
edges incident with the
vertex V.
Even and Odd Vertex: If
the degree of a vertex is
even, the vertex is called
an evenvertex and if the
degree of a vertex is odd,
the vertex is called an
odd vertex.
consider a graph below
177

Graphs – Basic concepts
Degree of a Graph: The degree of a graph is the
largest vertex degree of that graph. For the above
graph the degree of the graph is 3.
The Handshaking Lemma: In a graph, the sum of all
the degrees of all the vertices is equal to twice the
number of edges.
178

Types of Graphs
Null Graph
A null graph has no
edges. The null graph of
n vertices is denoted by
Nn
179

Types of graphs
Simple Graph
A graph is called simple
graph/strict graph if the
graph is undirected and
does not contain any
loops or multiple edges.
180

Types of graphs
Multi-Graph
If in a graph multiple
edges between the
same set of vertices are
allowed, it is called
Multigraph.
In other words, it is a
graph having at least
one loop or multiple
edges.
181

Types of graphs
Directed and Undirected
Graph
A graph G = (V, E) is called
a directed graph if the
edge set is made of
ordered vertex pair and a
graph is called undirected
if the edge set is made of
unordered vertex pair.
182

Types of graphs
Connected and
Disconnected Graph
A graph is connected if
any two vertices of the
graph are connected by
a path;.
183

Types of graphs
Disconnected graph: a
graph is disconnected if
at least two vertices of
the graph are not
connected by a path. If
a graph G is
disconnected, then
every maximal
connected subgraph of
G is called a connected
component of the graph
G.
184

Types of graphs
Regular Graph
A graph is regular if all
the vertices of the
graph have the same
degree. In a regular
graph G of degree r, the
degree of each vertex of
G is r.
185

Types of graphs
Complete Graph
A graph is called
complete graph if every
two vertices pair are
joined by exactly one
edge. The complete
graph with n vertices is
denoted by Kn
186

Types of graphs
Cycle Graph
If a graph consists of a
single cycle, it is called
cycle graph. The cycle
graph with n vertices
is denoted by Cn
187

Types of graphs
Bipartite Graph
If the vertex-set of a
graph G can be split into
two disjoint sets, 𝑉1 and
𝑉2, in such a way that
each edge in the graph
joins a vertex in 𝑉1 to a
vertex in 𝑉2, and there
are no edges in G that
connect two vertices in 𝑉
1 or two vertices in 𝑉2,
then the graph G is called
a bipartite graph.
188

Types of graphs
Complete Bipartite Graph
A complete bipartite
graph is a bipartite graph
in which each vertex in
the first set is joined to
every single vertex in the
second set. The complete
bipartite graph is denoted
by 𝐾𝑥,𝑦 where the graph
G contains x vertices in
the first set and y vertices
in the second set.
189

Representation of Graphs
There are mainly two ways to represent a graph:
1. Adjacency Matrix
2. Adjacency List
Adjacency Matrix
An Adjacency Matrix A[V][V] is a 2D array of size V×V
where V is the number of vertices in a undirected
graph. If there is an edge between Vx to Vy then the
value of A[Vx][ Vy]=1 and A[Vy][ Vx]=1, otherwise
the value will be zero. And for a directed graph, if
there is an edge between Vx to Vy, then the value of
A[Vx][ Vy]=1, otherwise the value will be zero.
190

Adjacency Matrix of an
Undirected Graph
Let us consider the
following undirected
graph and construct the
adjacency matrix:
191

Adjacency matrix
Adjacency matrix of the above undirected graph will
be:
a b c d
a 0 1 1 0
b 1 0 1 0
c 1 1 0 1
d 0 0 1 0
192

Adjacency Matrix of a Directed Graph
193

Adjacency Matrix of a Directed Graph
a b c d
a 0 1 1 0
b 0 0 1 0
c 0 0 0 1
d 0 0 0 0
194

Adjacency List
In adjacency list, an array
(A[V]) of linked lists is used
to represent the graph G
with V number of vertices.
An entry A[Vx] represents
the linked list of vertices
adjacent to the Vx-th
vertex. The adjacency list
of the undirected graph is
as shown in the figure:
195

Planar vs. Non-planar Graph
Planar graph: A graph G
is called a planar graph
if it can be drawn in a
plane without any
edges crossed. If we
draw graph in the plane
without edge crossing,
it is called embedding
the graph in the plane.
196

Non planar graph
Non-planar graph: A
graph is non-planar if it
cannot be drawn in a
plane without graph
edges crossing.
197

Isomorphism
If two graphs G and H contain the same number of
vertices connected in the same way, they are called
isomorphic graphs (denoted by G≅H).
It is easier to check non-isomorphism than
isomorphism. If any of these following conditions
occurs, then two graphs are non-isomorphic:
– The number of connected components are
different
– Vertex-set cardinalities are different
– Edge-set cardinalities are different
– Degree sequences are different
198

Isomorphic graphs
Example
The following graphs are isomorphic:
199

Homomorphism
A homomorphism from
a graph G to a graph H is
a mapping (May not be
a bijective mapping) h: G
H such that:
(x, y) ∈ E(G) (h(x), h(y))
∈ E(H) .It maps adjacent
vertices of graph G to
the adjacent vertices of
the graph H.
Properties of
Homomorphisms:
- A homomorphism is an
isomorphism if it is a
bijective mapping.
- Homomorphism always
preserves edges and
connectedness of a graph.
- The compositions of
homomorphisms are also
homomorphisms.
- To find out if there
exists any homomorphic
graph of another graph is
a Npcomplete problem. 200

Euler Graphs
A connected graph G is called an Euler graph, if
there is a closed trail which includes every edge of
the graph G. An Euler path is a path that uses every
edge of a graph exactly once.
An Euler path starts and ends at different vertices.
An Euler circuit is a circuit that uses every edge of a
graph exactly once. An Euler circuit always starts
and ends at the same vertex. A connected graph G
is an Euler graph if and only if all vertices of G are of
even degree, and a connected graph G is Eulerian if
and only if its edge set can be decomposed into
cycles.
201

Euler Graph
The above graph is an Euler graph as “a 1 b 2 c 3 d 4 e
5 c 6 f 7 g” covers all the edges of the graph.
202

Hamiltonian Graphs
A connected graph G is called Hamiltonian graph if there
is a cycle which includes every vertex of G and the cycle
is called Hamiltonian cycle. Hamiltonian walk in graph G
is a walk that passes through each vertex exactly once.
If G is a simple graph with n vertices, where n ≥ 3 If
deg(v) ≥ n/2 for each vertex v, then the graph G is
Hamiltonian graph. This is called Dirac's Theorem.
If G is a simple graph with n vertices, where n ≥ 2 if
deg(x) + deg(y) ≥ n for each pair of non-adjacent vertices
x and y, then the graph G is Hamiltonian graph. This is
called Ore's theorem.
204

Trees
Tree is a discrete structure that represents hierarchical
relationships between individual elements or nodes. A
tree in which a parent has no more than two children is
called a binary tree.
Tree and its Properties
Definition: A Tree is a connected acyclic undirected
graph. There is a unique path between every pair of
vertices in G. A tree with N number of vertices contains
(N-1) number of edges. The vertex which is of 0 degree
is called root of the tree. The vertex which is of 1
degree is called leaf node of the tree and the degree of
an internal node is at least 2.
206

Centers and Bi-Centers of a Tree
The center of a tree is a vertex with minimal
eccentricity. The eccentricity of a vertex X in a tree G
is the maximum distance between the vertex X and
any other vertex of the tree. The maximum
eccentricity is the tree diameter. If a tree has only
one center, it is called Central Tree and if a tree has
only more than one centers, it is called Bi-central
Tree. Every tree is either central or bi-central.
208

Algorithm to find centers and bi-centers
of a tree
Step 1: Remove all the vertices of degree 1 from the
given tree and also remove their incident edges.
Step 2: Repeat step 1 until either a single vertex or
two vertices joined by an edge is left. If a single
vertex is left then it is the center of the tree and if
two vertices joined by an edge is left then it is the bi-
center of the tree.
Example: Find out the center/bi-center of the
following tree:
209

centers and bi-centers of a tree
first, remove all
vertices of degree
1 and also remove
their incident
edges and get the
following tree:
210

Centers and bi-centers
Again, we will remove all vertices of degree 1 and also
remove their incident edges and get the following tree:
Finally we got a single vertex ‘c’ and we stop the
algorithm. As there is single vertex, this tree has one
center ‘c’ and the tree is a central tree.
211

Problem 2: Find out the center/bi-center of the
following tree:
212

Solution
At first, we will remove all vertices of degree 1 and
also remove their incident edges and get the
following tree:
213

Again, we will remove all vertices of degree 1 and also
remove their incident edges and get the following
tree:
Finally, we got two vertices ‘c’ and ‘d’ left, hence we
stop the algorithm. As two vertices joined by an edge is
left, this tree has bi-center ‘cd’ and the tree is bi-central.
214

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