MSc. Information Technology

Amity

University

Basic Mathematics Semester I

Preface

It gives me immense pleasure in bringing out the book "Basic Mathematics ", as the Mathematics subject is taught in most

Indian Engineering Institutions at the undergraduate level. The matter is represented in an easy way and covers particularly

the need of undergraduate students.

Any suggestions for improving the book will be highly appreciated.

Mrs.Archana Singh

Index

Subject Page no.

Chapter 1 SET THEORY

Chapter 2 MATHEMATICAL LOGIC

Chapter 3 MODERN ALGEBRA

Chapter 4 GRAPH THEORY

Chapter 5 DATA ANALYSIS

Chapter-I

SET THEORY

Contents:

1.1 Introduction to Set theory

1.2 Operation on Sets

1.3 Types of Sets

1.4 Venn Diagrams

1.5 Fundamental Laws of Set Operation

SET THEORY

1.1 Introduction to Set theory: Set theory lies at the foundation of all modern mathematics. It gives a very general framework in which

almost every branch of mathematics can be discussed.A set is a collection of objects, numbers, ideas,

etc. The different objects are called the elements or members of a set.

Each element of a set can be listed, or they may be represented using builder notation.

Give some examples of sets below.

V = {a, e, i, o, u}

A = {5, 10, 15, 20, ...}

C = {x | x N, 0 x 1000}

P is the set of all students in Math Studies.

Numerical Sets :So what does this have to do with math? When we define a set, all we have to specify

is a common characteristic. Who says we can't do so with numbers?

Set of even numbers: {..., -4, -2, 0, 2, 4, ...}

Set of odd numbers: {..., -3, -1, 1, 3, ...}

Set of prime numbers: {2, 3, 5, 7, 11, 13, 17, ...}

Positive multiples of 3 that are less than 10: {3, 6, 9}

And the list goes on. We can come up with all different types of sets.

There can also be sets of numbers that have no common trait, they are just defined that way. For

example:

{2, 3, 6, 828, 3839, 8827}

{4, 5, 6, 10, 21}

{2, 949, 48282, 42882959, 119484203}

Are all sets that I just randomly banged on my keyboard to produce.

Basic symbols:

, \in : belongs to

, \not\in : does not belong to

, @ : empty set

U , : universal set

, \subset : proper subset

, \not\subset : not a proper subset

, \subseteq : subset

, \not\subseteq : not a subset

, \cup : set union

Ai , \cup(i=1 to n) A_i : union of n sets

, \cap : set intersection

Ai , \cap(i=1 to n) A_i : intersection of n sets

, \bar A : complement of set A

(A) , P(A) : power set of set A

, X : Cartesian product

Ai , X(i=1 to n) A_i : cartesian product of n sets

1.2 Operation On Sets:

Union: The union of sets A and B, denoted by A U B , is the set defined as

A B = { x | x є A x є B }

Example 1: If A = {1, 2, 3} and B = {4, 5} , then A B = {1, 2, 3, 4, 5} .

Example 2: If A = {1, 2, 3} and B = {1, 2, 4, 5} , then A B = {1, 2, 3, 4, 5} .

Note that elements are not repeated in a set.

Intersection: The intersection of sets A and B, denoted by A B , is the set defined as

A B = { x | x є A x є B }

Example 3: If A = {1, 2, 3} and B = {1, 2, 4, 5} , then A B = {1, 2} .

Example 4: If A = {1, 2, 3} and B = {4, 5} , then A B = Ø

Difference: The difference of sets A from B , denoted by A - B , is the set defined as

A - B = { x | x є A x B }

Example 5: If A = {1, 2, 3} and B = {1, 2, 4, 5} , then A - B = {3} .

Example 6: If A = {1, 2, 3} and B = {4, 5}, then A - B = {1, 2, 3}.

Note that in general A - B ≠ B - A

Complement: For a set A, the difference U - A, where U is the universe, is called the complement of A

and it is denoted by .

Thus is the set of everything that is not in A.

Example 7: Suppose U = set of positive integers less than 10,

And A = {1, 2, 5, 6, 7}

Then, = {3, 4, 8, 9,}

The fourth set operation is the Cartesian product we first define an ordered pair and Cartesian product of

two sets using it. Then the Cartesian product of multiple sets is defined using the concept of n-tuple.

Ordered pair: An ordered pair is a pair of objects with an order associated with them.

An ordered pair <a, b > is defined in terms of sets as follows: <a, b > = {{a}, {a, b}}.

Similarly an ordered n-tuple can be defined as

<a1, a2, ..., an > = { { a1 }, { a1, a2 }, ... , { a1, a2, ..., an } }.

Two ordered pairs <a, b> and <c, d> are equal if and only if a = c and b = d. For example the ordered

pair <1, 2> is not equal to the ordered pair <2, 1>.

Cartesian product: The set of all ordered pairs <a, b>, where a is an element of A and b is an element

of B, is called the Cartesian product of A and B and is denoted by A×B .The concept of Cartesian

product can be extended to that of more than two sets. First we are going to define the concept of

ordered n-tuple.

Ordered n-tuple: An ordered n-tuple is a set of n objects with an order associated with them (rigorous

definition to be filled in). If n objects are represented by x1, x2, ..., xn, then we write the ordered n-tuple

as <x1, x2, ..., xn> .

Cartesian product: Let A1, ..., An be n sets. Then the set of all ordered n-tuples <x1, ..., xn> , where xi є

Ai for all i, 1 ≤ i ≤ n , is called the Cartesian product of A1, ..., An, and is denoted by A1×...×. An .

Equality of n-tuples: Two ordered n-tuples <x1, ..., xn> and <y1, ..., yn> are equal if and only if xi = yi

for all i, 1 ≤ i ≤ n .

For example the ordered 3-tuple <1, 2, 3> is not equal to the ordered n-tuple <2, 3, 1>.

1.3 Types Of Sets:

1.Finite Set –has a finite number of elements; you could list all elements in the set.

Example:

(a) A = {x:x is the river in India}

(b) C = {2, 4, 6, 8, .... ,1000000}

2.Infinite Set – has infinitely many members; you could not list all elements in the set

Example:

(a) Z = {-1, -2, -3, -4, ... }

(b) B = {all real numbers between 2 and 5}

Hint: Infinite sets are those which include all R, Q, or Q’ between two values or end with ...

3.Null or Empty Set – the set with no elements; { } or

Example:

D = {x:x 2 = 5 and x is an integer}

= , since there is no integer whose square is 5

The null set is a subset of every set, including the null set itself.

4.Equality of Set – Two sets are equal if they have precisely the same members. Now, at first glance they may not seem equal, you may have to examine them closely!

And the equals sign (=) is used to show equality, so we would write:

A = B

More formally, for any sets A and B, A = B if and only if x [ x є A x є B ] .

Thus for example {1, 2, 3} = {3, 2, 1}, that is the order of elements does not matter, and {1, 2, 3} = {3,

2, 1, 1}, that is duplications do not make any difference for sets.

5. Equivalent Sets–Two sets are equivalent if it is possible to pair off members of the first set with

members of the second, with no leftover members on either side. To capture this idea in set-theoretic

terms, the set A is defined as equivalent to the set B (symbolized by A ≡ B) if and only if there exists a

third set the members of which are ordered pairs such that: (1) the first member of each pair is an

element of A and the second is an element of B, and (2) each member of A occurs as a first member and

each member of B occurs as a second member of exactly one pair. Thus, if A and B are finite and A ≡ B,

then the third set that establishes this fact provides a pairing, or matching, of the elements of A with

those of B. Conversely, if it is possible to match the elements of A with those of B, then A ≡ B, because a

set of pairs meeting requirements (1) and (2) can be formed

Example: Let A= {a, b, c, d} and B= {1, 2, 3, 4} be two sets. Clearly A is not equal to b. However, the

elements of A can be put into one-to-one correspondence with those of B, therefore we write A ≡ B.

6.Subset –If every member of set A is also a member of set B, then A is said to be a subset of B, written

A B (also pronounced A is contained in B). Equivalently, we can read as B is a superset of A, B

includes A, or B contains A. The relationship between sets established by is called inclusion or

containment.

If all the members of a set M are also members of a set P, then M is a subset of P: M P .

*Note: M could be exactly the same as P.

Example

(a) If A 1,2 and

B 2, 1,0,1,2 and

C 2, 1,0,1,2 then.

A B

A C

C B

A A

(b) Every set has at least two subsets, itself and the null set.

List all the subsets of

a, b, c . Use proper notation

7.Proper Subsets– If all the members of a set M are also members of a set P and M is a smaller set

than P, then M is a proper subset of P: M P .

Example: {1, 2, 3} is a subset of {1, 2, 3}, but is not a proper subset of {1, 2, 3}.On the contrary, {1, 2,

3} is a proper subset of {1, 2, 3, 4} because the element 4 is not in the first set.

8. Power set– The set of all subsets of a set A is called the power set of A and denoted by 2A or (A).

Example:

(a) A = {1, 2}, (A) = {Ø, {1}, {2}, {1, 2} } .

(b) B = {{1, 2}, {{1}, 2}, Ø } , (B) = { Ø, {{1, 2}}, {{{1}, 2}}, { Ø }, { {1, 2}, {{1}, 2 }}, { {1,

2}, Ø }, { {{1}, 2}, Ø }, {{1, 2}, {{1}, 2}, Ø } } .

9.Universal Set–The set which contains all the available elements for a particular problem.

The complement of a set A is defined to be the set of all elements of the universal set which are not in A.

Note that A U Ac is always the universal set, while

A Ac = Ø

The set U is the superset of every set

10. Family of Sets– a collection F of subsets of a given set S is called a family of subsets of S, or a

family of sets over S. More generally, a collection of any sets whatsoever is called a family of sets.

Example: If A = {1, 2} then the set {Ø, {1}, {2}, {1, 2}} is the family of sets whose elements are

subsets of the set A.

1.4 Venn Diagrams:

Venn diagrams are used to represent sets. Here, the set A{1, 2, 4, 8} is shown using a circle. In Venn

diagrams, sets are usually represented using circles. The universal set is the rectangle. The set A is a

subset of the universal set and so it is within the rectangle.

The complement of A, written Ac, contains all events in the sample space which are not members of A.

A and Ac together cover every possible eventuality.

A B means the union of sets A and B and contains all of the elements of both A and B. This can be

represented on a Venn Diagram as follows:

A B means the intersection of sets A and B. This contains all of the elements which are in both A and

B. A B is shown on the Venn Diagram below:

An important result connecting the number of members in sets and their unions and intersections is:

n(A) + n(B) - n(A B) = n(A B)

1.5 Fundamental Laws of Set Operation :

i. Identity Laws

A U Ø = A

A U = A

ii. Domination Laws

A U U = U

A Ø = Ø

iii. Idempotent Laws

A U A = A

A A = A

iv. Commutative Laws

A U B = B U A

A B = B A

v. Associative Laws

(A U B) U C = A U (B U C)

(A B) C

= A (B

C)

vi. Distributive Laws

A U (B C) = (A U B) (A U C)

A (B U C) = (A B) U (A C)

Proof: x ε A U (B ∩ C)

x ε A or x ε (B ∩ C)

x ε A or (x ε B and x ε C)

(x ε A or x ε B) and (x ε A or x ε C)

x ε (A U B) and x ε (A U C)

x ε (A U B) ∩ (A U C)

Hence: A U (B ∩ C) = (A U B) ∩ (A U C)

Similarly, the second distributive law can also be proved.

vii. De Morgan's Laws

a. Complement of the intersection of two sets is the union of their complements, i.e.

Proof:

or

or

Therefore

b. Complement of the unions of two sets is the intersection of their complements, i.e.

Proof: same as above

Chapter-I

SET THEORY

End Chapter quizzes :I

Ques 1. The freshman science classes were surveyed to see whether they wanted to visit the local natural

history museum or a nearby state park. They could choose the museum, the park, both, or neither. The

results of the survey are shown in the Venn diagram at the right. If 96 students were surveyed, how

many wanted to go only to the state park?

a. 9

b. 10

c. 11

d. 12

Ques 2. A = {x is an even number, x > 1}

B = {x is an odd number, x > 1}; then A U B is defined as

a. A U B ={ x is an even number, x > 1}

b. A U B ={x is an odd number, x > 1}

c. A U B = {2,3,4,5,6,…. }

d. A U B ={ 0,1,2,3,4,5,6,…}

Ques 3. The intersection of the sets {1, 2, 3} and {2, 3, 4} is

a. {2, 3}.

b. {1,2, 3,4}.

c. {3}.

d. {1,2,2,3,3}.

Ques 4. A = {1,2,3}and B = {3,4,5}; then A - B is defined as

a. {2, 3}.

b. {1,2, 3,4}.

c. {3}.

d. {1,2, 4,5}.

Ques 5. If A and B are any two sets and

A U B = A ∩ B

then

a. A=B

b. A≤B

c. A B

d. A U B = Ø

Ques 6. If the intersection of two sets A and B is empty, the two sets are said to be

a. Universal set

b. Disjoint set

c. Finite set

d. Infinite set

Ques 7. According to De Morgan's Laws; Complement of the intersection of two sets is the union of

their complements, i.e.

a.

b. c. A U B = A ∩ B

d. None

Ques 8. According to the Identity Laws,which one is correct

a. A U Ø = Ac

b. A U Ø = U

c. A U Ø = Ø

d. A U Ø = A

Ques 9. A U Ac is always the universal set

a. True

b. False

c. Depends on the set

d. None

Ques 10. (Ac) c

= A

a. True

b. False

c. Depends on the set

d. None

Chapter-II

MATHEMATICAL LOGIC

Contents:

2.1 Basic concepts of Mathematical logic

2.2 Propositions or Statements

2.3 Combining Propositions

2.4 Truth Table

2.5 Conditional and Biconditional Propositions

2.6 Reverse, Converse, Inverse, and Contrapositive of an implication

2.7 Tautology

2.8 Logical Equivalence

2.9 Switching circuits

MATHEMATICAL LOGIC

2.1 Basic concepts of Mathematical logic: Mathematical logic is the application of mathematical

techniques to logic. Logic is commonly known as the science of reasoning. The emphasis here will be

on logic as a working tool. We will develop some of the symbolic techniques required for computer

logic. Some of the reasons to study logic are the following:

At the hardware level the design of ’logic’ circuits to implement instructions is greatly simplified

by the use of symbolic logic.

At the software level a knowledge of symbolic logic is helpful in the design of programs.

Logic is a language for reasoning. It is a collection of rules we use when doing logical reasoning. Human reasoning has been observed over centuries from at least the times of Greeks, and patterns appearing in reasoning have been extracted, abstracted, and streamlined. The foundation of the logic we are going to learn here was laid down by a British mathematician George Boole in the middle of the 19th century, and it was further developed and used in an attempt to derive all of mathematics by Gottlob Frege, a German mathematician, towards the end of the 19th century. A British philosopher/mathematician, Bertrand Russell, found a flaw in basic assumptions in Frege’s attempt but he, together with Alfred Whitehead, developed Frege’s work further and repaired the damage. The logic we study today is more or less along this line. In logic we are interested in true or false of statements, and how the truth/falsehood of a statement can be determined from other statements. However, instead of dealing with individual specific statements, we are going to use symbols to represent arbitrary statements so that the results can be used in many similar but different cases. The formalization also promotes the clarity of thought and eliminates mistakes. There are various types of logic such as logic of sentences (propositional logic), logic of objects (predicate logic), logic involving uncertainties, logic dealing with fuzziness, temporal logic etc. Here we are going to be concerned with propositional logic and predicate logic, which are fundamental to all types of logic.

2.2 Propositions or Statements:

The smallest unit we deal with in propositional logic is a sentence. We do not go inside individual sentences and analyze or discuss their meanings. We are going to be interested only in true or false of sentences, and major concern is whether or not the truth or falsehood of a certain sentence follows from those of a set of sentences, and if so, how. Propositional logic is a logic at the sentential level. Thus sentences considered in this logic are not arbitrary sentences but are the ones that are true or false. This kind of sentences is called propositions. If a proposition is true, then we say it has a truth value of ―true‖; if a proposition is false, its truth value is ―false‖. A proposition is any meaningful statement that is either true or false, but not both e.g.

p : 1 + 1 = 3

to define p to be the proposition 1 + 1 = 3: The truth value of a proposition is true, denoted by T, if it is a true statement and false, denoted by F, if it is a false statement. Statements that are not propositions include questions and commands.

2.3 Combining Propositions:

Simple sentences which are true or false are called basic propositions. Larger and more complex sentences are constructed from basic propositions by combining them with connectives. Thus propositions and connectives are the basic elements of prepositional logic.

In English, we can modify, combine, and relate propositions with words such as ―not‖,―and‖,

―or‖, ―implies‖, and ―if then‖.For example, we can combine three propositions into one like this:

If all humans are mortal and all Indians are human, then all Indians are mortal

The propositions and connectives are the basic elements of prepositional logic. A few of connectives are as follows:

Mathematics Symbol Meaning

or ¬ not

and

or

implies

if and only if

exclusive or

p truth variable

T true

F false

Example

Consider the following propositions

p : It is Friday:

q : It is raining:

Construct the propositions p ^ q and p v q:

Solution.

The conjunction of the propositions p and q is the proposition

p ^ q : It is Friday and it is raining:

The disjunction of the propositions p and q is the proposition

p v q : It is Friday or It is raining:

A truth table displays the relationships between the truth values of propositions.

Next, we display the truth tables of p ^ q and p v q:

p q p ^ q

Let p and q be two propositions. The exclusive of p and q; denoted p © q;

is the proposition that is true when exactly one of p and q is true and is false

otherwise. The truth table of the exclusive ’or’ is displayed below

2.4 Truth Table :

A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra,

Boolean function and propositional calculus—to compute the functional values of logical expressions

on each of their functional arguments

Example: Constructing a Truth Table

Construct the truth table for ~ (p q).

Solution: Whenever we encounter a complex formula like this, we work from the inside out, just as we

might do if we had to evaluate an algebraic expression, like - (a+b). Thus, we start with the p and q

columns, then construct the p q column, and finally, the ~ (p q) column:

p q p q ~(p q)

T T T F

T F F T

F T F T

F F F T

2.5 Conditional and Biconditional Propositions :

Let p and q be propositions. The implication p q is the the proposition that is false only when p is true

and q is false; otherwise it is true. p is called the hypothesis and q is called the conclusion. The

connective is called the conditional connective.

Example:

Construct the truth table of the implication p q:

Solution

The truth table is

2.6 Reverse, Converse, Inverse, and Contrapositive of an implication

By definition, the reverse of an implication means the same as the original implication itself. Each

implication implies its contrapositive, even intuitionistically. In classical logic, an implication is

logically equivalent to its contrapositive, and, moreover, its inverse is logically equivalent to its

converse.

Comparisons:

name form description

implication if P then Q first statement implies truth of second

inverse if not P then not Q negation of both statements

converse if Q then P reversal of both statements

contrapositive if not Q then not P reversal of negation of both statements

Take the statement "All quadrilaterals have four sides," or equivalently expressed "If a shape is a

quadrilateral, then it has four sides."

The Contrapositive is "If a shape does not have four sides, then it is not a quadrilateral." This

follows logically, and as a rule, contrapositives share the truth value of their conditional.

The Inverse is "If a shape is not a quadrilateral, then it does not have four sides." In this case,

unlike the last example, the inverse of the argument is true.

The Converse is "If a shape has four sides, then it is a quadrilateral." Again, in this case, unlike

the last example, the converse of the argument is true.

The Contradiction is "There is at least one quadrilateral that does not have four sides."

Since the statement and the converse are both true, it is called a biconditional, and can be expressed as

"A shape is a quadrilateral if, and only if, it has four sides." That is, having four sides is both necessary

to be a quadrilateral, and alone sufficient to deem it a quadrilateral.

2.7 Tautology :

A formula of propositional logic is a tautology if the formula itself is always true regardless of which

valuation is used for the propositional variables.

There are infinitely many tautologies. Examples include:

("A or not-A"), the law of the excluded middle. This formula has only one

propositional variable, A. Any valuation for this formula must, by definition, assign A one of the

truth values true or false, and assign A the other truth value.

("if A implies B then not-B implies not-A", and vice versa), which expresses

the law of Contraposition.

("if not-A implies both B and its negation not-B, then not-A

must be false, then A must be true"), which is the principle know as reduction ad absurdum.

, which is know as de Morgan’s law

("if A implies B and B implies C, then A implies C"), which is the

principle know as syllogism.

(if A or B is true and both implies C, then C must be true),

which is the principle know as proof by cases..

The problem of determining whether a formula is a tautology is fundamental in propositional logic. The

definition suggests one method: proceed by cases and verify that every possible valuation does satisfy

the formula. An algorithmic method of verifying that every valuation causes this sentence to be true is to

make a truth table that includes every possible valuation.

For example, consider the formula

There are 8 possible valuations for the propositional variables A, B, C, represented by the first three

columns of the following table. The remaining columns show the truth of subformulas of the formula

above, culminating in a column showing the truth value of the original formula under each valuation.

A B C

T T T T T T T T

T T F T F F F T

T F T F T T T T

T F F F T T T T

F T T F T T T T

F T F F T F T T

F F T F T T T T

F F F F T T T T

Because each row of the final column shows T, the sentence in question is verified to be a tautology.

Example:

Example:

2.8 Logical Equivalence:

Two propositions whose truth tables have the same last column are called logically equivalent. This

means that no matter what truth values the primitive propositions have, these two propositions are either

both true or both false. To test whether or not two propositions are logically equivalent we make a truth

table for each of them and compare their last columns.

Example: The proposition (p ^ q) ^ r is logically equivalent to p ^ (q ^ r). This is why the expression p ^

q ^ r is invalid, wherever the parentheses go the result is equivalent.

p q r

p ^ q (p ^ q) ^ r

T T T T T

T T F F F

T F T F F

T F F F F

F T T F F

F T F F F

F F T F F

F F F F F

p q r

q ^ r p^ (q ^ r)

T T T T T

T T F F F

T F T F F

T F F F F

F T T F F

F T F F F

F F T F F

F F F F F

Similarly, the proposition (p ^ q) ^ r is logically equivalent to p ^ (q ^ r) so the expression p ^ q ^ r is

invalid. Wherever the parentheses go the result is equivalent.

De Morgan's Laws state that (p ^ q) is logically equivalent to p V q and that

(p V q) is logically equivalent to p ^ q.

2.9 Switching Circuits:

A switching network is an arrangement of wires and switches which connect two terminals. A switch can be either closed or open. A closed switch permits and an open switch prevents flow of current.

The simplest kind of network in which there is a single wire containing a single switch p is shown in the

figure:

P

If P denotes a switch, then p' denotes that switch which is open when p is closed, and is closed when p

is open.

Let P be the statement: "switch p is closed", then lp will be the statement: "switch p is open." If x denotes the state of the switch p, then x' will be the state of the switch p', x will be called the state variable or Boolean variable. It is a binary variable.

If the value x=l denotes that the switch is closed or the current flows. Then value x=O denotes that the switch is open or the current stops. , Switch p

State of p Boolean Proposition

al

Truth

Value Value of x

Variable Variable

Closed

(on) x P T 1

Open

(off) x ⌐P F 0

Let 'in series' connection be represented by /\ [i.e. a/\b denotes' switches a and bare connected in series']. Also let aV b denote 'switches a and b are connected in parallel'.

Consider the system ({O,l} /\, V ); B={O,1}

Then B is a non-empty set and /\ and V are two binary compositions on B, by tables: /\ 0 1

0 0 0

1 0 1

v 0 1

0 0 1

1 1 1 The conditions of idempotency, commutativity, associativity and absorption are clearly seen to be satisfied,

Hence B is a distributive lattice in which each element has a complement, i.e., it is a Boolean algebra.

Thus the system ({O,l} /\, V, `) is usually called switching algebra.

Representation of Circuits We now illustrate how a given Boolean function can represent a circuit and how a circuit can be represented by a Boolean function.

Simplification of Circuits Simplification of a circuit would normally mean the least complicated circuit with minimum cost and best (convenient) results. This would be governed by various factors like cost of equipment, positioning and number of switches, type of material used etc. Simplification of circuits would mean lesser number of switches, which we achieve by qsing different properties of Boolean algebras.

Example

Draw the circuit representing the Boolean functions: (i) a/\b,

(ii) aVb,

(iii) a/\(bVc) .

Solution

Example:

Find the Boolean function representing the circuit:

Solution

The Boolean function for the given circuit is

a/\[(b/\c)v(d/\(e/\f))]

Design of Circuits Here we explain with the help of examples how with given requirements, corresponding circuits are designed.

Example

In a committee of three members, each member indicates his agreement by closing a switch. A bulb is lighted when a majority agrees. Construct a suitable arrangement.

Solution

Let A, B, C denote agreement by member number 1, 2, 3

respectively.

The bulb is lighted when current flows. Current flows when at least two members agree. Hence the logic statement describing agreement is

f = (A/\B/\C)V(A/\B/\¬C)V(A/\⌐B/\C)V(¬A/\B/\C) ........................ (i)

The circuit diagram- for the Boolean function: [here a stands for A, b stands

for B and c stands for C]

Chapter-II

MATHEMATICAL LOGIC

End Chapter quizzes :II

Ques 1.Which of the following are propositions? Give the truth value of the propositions. a. The difference of two primes. b. 2 + 2 = 4: c. Washington D.C. is the capital of New York. d. How are you? Solution . a. Not a proposition. b. A proposition with truth value (T). c. A proposition with truth value (F). d. Not a proposition.

Ques 2.The symbol of if and only if

a. b. c.

d. ¬

Ques 3. p ^ q implies a. p and q b. p or q

c. exclusive of p and q

d. p not q

Ques 4. For two statements P and Q, implication implies

a. if not P then not Q

b. if Q then P

c. if not Q then not P

d. if P then Q

Ques 5. If a formula itself is always true regardless of which valuation is used for the

propositional variables,the propositional logic is called

a. Tautology

b. Biconditional Propositions

c. Combining Propositions

d. Contrapositive of an implication

Ques 6. , means

a. A or not-A

b. if A implies B then not-B implies not-A

c. if not-A implies both B and its negation not-B, then not-A must be false, then A must be

true

d. if A implies B and B implies C, then A implies C

Ques 7. De Morgan's Laws state that (p ^ q) is logically equivalent to p V q a. True b. False c. Not always true d. Depends on the variable value

Ques 8. p: "There is life on Mars."

q: "There is life on Jupiter."

~ (p q) is

a. There is no life on Mars and there is no life on Jupiter

b. There is no life on Mars and there is life on Jupiter

c. There is neither life on Mars nor on Jupiter

d. There is life on atmost one of Mars or Jupiter

Ques 9.For the implication formula A →B ; ¬ A→ ¬ B is

a. Reverse

b. Inverse

c. Contrapositive

d. Converse Ques 10.Which one of them is not dealt in the topic

a. Venn Diagram b. Logical Equivalence c. Truth table

d. Contrapositive

Chapter-III

MODERN ALGEBRA

Contents:

3.1 Binary Operations

3.2 Properties of Binary operations

3.3 Semigroup

3.4 Monoid

3.5 Group

3.6 Groupoid

MODERN ALGEBRA

3.1 Binary Operations: A binary operation is simply a rule for combining two objects of a given

type, to obtain another object of that type. Through elementary school and most of high school,

the objects are numbers, and the rule for combining numbers is addition, subtraction,

multiplication or division.

Binary operation on a set S. A binary operation on a set S is a rule which assigns to each ordered

pair a,b of elements in S a unique element c = ab.

Closure. A set S is closed with respect to a binary operation if and only if every image ab is in S

for every a,b in S.

3.2 Properties of Binary operations:

Commutative operation: A binary operation on a set S is called commutative if

xy = yx for all x,y in S.

Associative operation: A binary operation on a set S is called associative if

(xy)z = x (yz) for all x,y,z in S.

Distributive: Let S be a set on which two operations ∙ and + are defined. The operation ∙ is said to

left distributive with respect to + if

a ∙(b + c ) = (a∙b) + (a∙c) for all a,b,c in S

and is said to be right distributive with respect to + if

(b + c)∙a = (b∙a) + (c∙a) for all a,b,c in S

Existence of identity elements and inverse elements:

Identity element: A set S is said to have an identity element with respect to a binary operation

on S if there exists an element e in S with the property ex = xe = x for every x in S.

Inverse element: If a set S contains an identity element e for the binary operation , then an

element b S is an inverse of an element a S with respect to if ab = ba = e .

Note. There must be an identity element in order for inverse elements to exist.

Theorems :

Theorem 1. A set S contains at most one identity for the binary operation . An element e is called

a left identity if ea = a for every a in S. It is called a right identity if ae = a for every a in S. If a set

contains both a left and a right identity, they are the same.

Theorem 2. An element of a set S can have at most one inverse if the operation is associative.

In general, in regular algebra, when one multiplies several real numbers together, a product of

several numbers is assumed to have a particular value independent of how the

multiplications are performed (i.e. where parentheses are placed):

x1x2x3 ... xn = x1(x2x3)(x4 ... xn) = (x1x2)(x3x4)(x5 ...xn)

or, in terms of numbers,

5∙3∙8∙7∙3∙9 = 5(3∙8)(7∙3)9 = (5∙3)(8∙7)(3∙9) = ...

The product is unique, independent of the placing of the parentheses. This rule is true in the case of

the multiplication of real numbers. It is not, however, in general true with an arbitrary operation .

Under what conditions is it true? It is true on a closed set S which has an operation which is

associative. The operation of multiplication on the real numbers is associative and so this product

is unique for the multiplication of real numbers.

Theorem 3. Let a set S be closed with respect to an associative binary operation . Then the

products formed from the factors , multiplied in that order, and with the parentheses

placed in any positions whatever, are equal to the general product .

Note that the theorem refers to the grouping -- the order of the numbers remains the same.

The concept of a binary operation is a very general one, and need not be restricted to sets of

numbers. In fact, an operation can be specified on any finite set simply by presenting a table that

shows how the operation is performed, when you are given two elements of the set. For example,

consider the set and an operation, denoted by *, defined by the following table:

We interpret this operation table in much the same way that we would interpret an addition table.

Using the operation symbol * as we would use + to mean addition, the table shows us, among

other things, that

and so on. The table summarizes 16 such calculations, telling us how to combine each of the four

elements of A with each of the four elements.

Not All Operations Have the Same Properties

You should be familiar with various properties of the arithmetic operations on numbers. Addition

of numbers, for instance, is a commutative operation -- meaning that for all numbers

x and y. The operation on the set A defined by the operation table above, however, is not

commutative, and there are several instances of this lack of commutativity. For

instance, since the table shows that . In general,

commutativity is a property of an operation, so it takes only one instance of lack of commutativity

to spoil that property for the operation. It is easy to check whether an operation defined by a table

is commutative. Simply draw the diagonal line from upper left to lower right, and then look to see

if the table is symmetric about this line. In the illustration below, we see a lack of symmetry: the

table entries colored yellow do not match, and the table entries colored blue do not match.

Either one of these mismatches would be sufficient to make the operation non-commutative.

Addition of numbers is an associative operation, meaning that for all

numbers x, y and z. To check to see whether the operation * defined above is associative,

however, is a somewhat tedious task. We would need to compute all combinations of the

form in two ways -- once as shown, and then again in the form -- and

then check to see that they are equal. This must be done for each selection of elements to fill the

placeholders. In the case of a 4-element set such as A above, there are choices of the

elements to be used, and each must be computed in two ways. Thus, to verify that a binary

operation on a 4-element set is associative, we would have to do 128 computations! There is no

easy shortcut as there is for checking commutativity.

On the other hand, if a given operation fails to be associative, all we need to do to verify this is to

find one instance of the lack of associativity. For the operation * defined on the set

above, we find that , while . Thus ,

so the operation * is not associative

3.3 Semigroup:

A mathematical object defined for a set and a binary operator in which the multiplication operation

is associative. No other restrictions are placed on a semigroup; thus a semigroup need not have an

identity element and its elements need not have inverses within the semigroup. A semigroup is an

associative groupoid. A semigroup with an identity is called a monoid.

A semigroup can be empty.

3.4 Monoid:

A monoid is a set that is closed under an associative binary operation and has an identity element

such that for all , . Note that unlike a group, its elements need not have

inverses. It can also be thought of as a semigroup with an identity element.

A monoid must contain at least one element. A monoid that is commutative is, not surprisingly,

known as a commutative monoid.

3.5 Group:

A group G is a finite or infinite set of elements together with a binary operation (called the group

operation) that together satisfy the four fundamental properties of closure, associativity, the

identity property, and the inverse property. The operation with respect to which a group is defined

is often called the "group operation," and a set is said to be a group "under" this operation.

Elements A, B, C, ... with binary operation between A and B denoted AB form a group if

1. Closure: If A and B are two elements in G, then the product AB is also in G.

2. Associativity: The defined multiplication is associative, i.e., for all , .

3. Identity: There is an identity element I (a.k.a. 1, , or ) such that for every element

.

4. Inverse: There must be an inverse (a.k.a. reciprocal) of each element. Therefore, for each

element A of G , the set contains an element such that .

A group is a monoid each of whose elements is invertible.

A group must contain at least one element, with the unique single-element group known as the

trivial group.

The study of groups is known as group theory. If there are a finite number of elements, the group is

called a finite group and the number of elements is called the group order of the group. A subset of

a group that is closed under the group operation and the inverse operation is called a subgroup.

Subgroups are also groups and many commonly encountered groups are in fact special subgroups

of some more general larger group.

A basic example of a finite group is the symmetric group Sn, which is the group of permutations

(or "under permutation") of n objects. The simplest infinite group is the set of integers under usual

addition. For continuous groups, one can consider the real numbers or the set of n X n invertible

matrices. These last two are examples of Lie groups.

3.6 Groupoid:

There are at least two definitions of "groupoid" currently in use.

The first type of groupoid is an algebraic structure on a set with a binary operator. The only

restriction on the operator is closure (i.e., applying the binary operator to two elements of a given

set S returns a value which is itself a member of S ). Associativity, commutativity, etc., are not

required. A groupoid can be empty. An associative groupoid is called a semigroup.

The second type of groupoid is an algebraic structure first defined by Brandt (1926) and also

known as a virtual group. A groupoid with base B (or "over B ") is a set Gwith mappings α and β

from G onto B and a partially defined binary operation , satisfying the following four

conditions:

1. is defined whenever , and in this case and .

2. Associativity: if either of and are defined so is the other and they are equal.

3. For each , there are left- and right-identity elements and respectively, satisfying

.

4. Each has an inverse satisfying and .

Any group is a groupoid with base a single point.

The most basic example of groupoid with base is the pair groupoid, where , and

, , and with multiplication . Any equivalence relation on

defines a subgroupoid of the pair groupoid.

A useful way to think of a groupoid is as a parametrized equivalence relation on , as follows.

Given a groupoid over , define an equivalence relation on by for each . This

equivalence relation is "parameterized" because there may be more than one element in which

give rise to the same equivalence, that is, and such that and .

Chapter-III

MODERN ALGEBRA

End Chapter quizzes :III

Ques. 1 The greatest common divisor of 42 and 60 equals to

a. 6

b. 4

c. 8

d. 3

Ques.2 If ea = a for every a in S, then e is called

a. right identity

b. left identity

c. right inverse

d. left inverse

Ques.3 There must be an identity element in order for inverse elements to exist a. Always true b. False c. Depends upon the elements of the set d. None

Ques.4 An algebric structure (G,*), satisfying only the closure property and the associative law, is called

a. Semigroup

b. Monoid

c. Group

d. Groupoid

Ques.5 A monoid each of whose elements is invertible, is called

a. Semigroup

b. Cyclic group

c. Group

d. Groupoid

Ques.6 Let S be a set on which two operations ∙ and + are defined. The operation ∙ is said to left

distributive with respect to + if for all a,b,c in S

a. a ∙(b + c ) = (b + c)∙a

b. (b + c)∙a = b + c + a

c. a ∙(b + c ) = (a∙b) + (a∙c)

d. (b + c)∙a = (b∙a) + (c∙a)

Ques 7 A semigroup with an identity element, is called

a. Cyclic group

b. Monoid

c. Group

d. Groupoid

Ques.8 Which one of the following is true

a. A group must contain at least one element

b. A monoid must contain at least one element

c. A semigroup can be empty

d. All are true

Ques 9 An associative groupoid is called a

a. Cyclic group

b. Monoid

c. Group

d. Semigroup

Ques.10 A subset of a group that is closed under the group operation and the inverse operation is

called a subgroup

a. Cyclic group

b. Subgroup

c. Abelian group

d. Semigroup

Chapter-IV

Graph Theory

Contents:

4.1 Graph

4.2 Multigraph

4.3 Complete graph

4.4 Bi Graph/Bipartite Graph

4.5 Degree

4.6 Degree Sequence

Graph Theory

4.1 Graph : In mathematics a graph is an abstract representation of a set of objects where some

pairs of the objects are connected by links. The interconnected objects are represented by

mathematical abstractions called vertices, and the links that connect some pairs of vertices are

called edges. Typically, a graph is depicted in diagrammatic form as a set of dots for the vertices,

joined by lines or curves for the edges.

A graph is an ordered pair G: = (V, E) comprising a set V of vertices or nodes together with a set E

of edges or lines, which are 2-element subsets of V. To avoid ambiguity, this type of graph may be

described precisely as undirected and simple.An undirected graph G consists of a set VG of vertices

and a set EG of edges such that each edge e 2 EG is associated with an unordered pair of vertices,

called its endpoints. A directed graph or digraph G consists of a set VG of vertices and a set EG of

edges such that each edge e 2 EG is associated with an ordered pair of vertices. We denote a graph

by G = (VG; EG):Two vertices are said to be adjacent if there is an edge connecting the two

vertices. Two edges associated to the same vertices are called parallel. An edge incident to a single

vertex is called a loop. A loop is an edge (directed or undirected) which starts and ends on the

same vertex; these may be permitted or not permitted according to the application. In this context,

an edge with two different ends is called a link.

A vertex that is not incident on any edge is called an isolated vertex. A graph with neither loops

nor parallel edges is called simple graph. If multiple edges are allowed between vertices, the graph

is known as a multigraph. Vertices are usually not allowed to be self-connected, but this restriction

is sometimes relaxed to allow such "graph loops." A graph that may contain multi edges and graph

loops is called a pseudo graph.

Example Consider the following graph G

a. Find EG and VG. b. List the isolated vertices. c. List the loops. d. List the parallel edges. e. List the vertices adjacent to v3:. f. Find all edges incident on v4: Solution. a. EG = fe1; e2; e3; e4; e5; e6g and VG = fv1; v2; v3; v4; v5; v6; v7g: b. There is only one isolated vertex, v5: c. There is only one loop, e5: d. fe2; e3g: e. fv2; v4g: f. fe1; e4; e5g: Example Which one of the following graphs is simple.

Solution. a. is not simple since it has a loop and parallel edges. b. is simple. A complete graph on n vertices, denoted by Kn; is the simple graph that contains exactly one edge between each pair of distinct vertices.

4.2 Multigraph:

The term "multigraph" is generally understood to mean that multiple edges (and sometimes loops)

are allowed. Some references require that multigraphs possess no graph loops, some explicitly

allow them whereas there are others who uses the term "multigraph" to mean a graph containing

either loops or multiple edges.

Where graphs are defined so as to allow loops and multiple edges, a multigraph is often defined to

mean a graph without loops, however, where graphs are defined so as to disallow loops and

multiple edges, the term is often defined to mean a "graph" which can have both multiple edges

and loops, although many use the term "pseudograph" for this meaning

4.3 Complete graph:

A complete graph is a graph in which each pair of graph vertices is connected by an edge. The

complete graph with n graph vertices is denoted Kn and has (the triangular

numbers) undirected edges, where is a binomial coefficient.The complete graph on n vertices

has n vertices and n(n-1)/2 edges, and is denoted by Kn. It is a regular graph of degree n − 1. All

complete graphs are their own cliques. They are maximally connected as the only vertex cut (A

vertex cut of a connected graph G is a set of vertices whose removal renders G disconnected)

which disconnects the graph is the complete set of vertices.In older literature, complete graphs are

sometimes called universal graphs.

A complete graph with n nodes represents the edges of an (n-1)-simplex. Geometrically K3

relates to a triangle, K4 a tetrahedron, K5 a pentachodron,

etc.

4.4 Bi Graph/Bipartite Graph:

A bigraph graph, also called a bipartite, is a set of graph vertices decomposed into two disjoint sets

such that no two graph vertices within the same set are adjacent. A bipartite graph is a special case

of a k-partite graph with k=2. Bi graphs are equivalent to two-colorable graphs, and a graph is

bipartite iff all its cycles are of even length

U V

The two sets U and V may be thought of as a coloring of the graph with two colors: if we color all

nodes in U blue, and all nodes in V green, each edge has endpoints of differing colors, as is

required in the graph coloring problem. In contrast, such a coloring is impossible in the case of a

nonbipartite graph, such as a triangle: after one node is colored blue and another green, the third

vertex of the triangle is connected to vertices of both colors, preventing it from being assigned

either color.

All trees are bipartite, e.g.

4.5 Degree :

The degree (or valency) of a vertex of a graph is the number of edges incident to the vertex, with

loops counted twice. The degree of a vertex v is denoted deg (v). The maximum degree of a graph

G, denoted by Δ(G), is the maximum degree of its vertices, and the minimum degree of a graph,

denoted by δ(G), is the minimum degree of its vertices.

. In the above graph, the maximum degree is 3 and the minimum degree is 0.

3

3

3

2

2

1

1

0

In this graph all of the vertices have degree three.

4.6 Degree Sequence:

Given an undirected graph, a degree sequence is a monotonic non increasing sequence of the

vertex degrees (valencies) of its graph vertices. The number of degree sequences for a graph of a

given order is closely related to graphical partitions. The sum of the elements of a degree sequence

of a graph is always even due to fact that each edge connects two vertices and is thus counted

twice .For the above graph degree sequence is (3, 3, 3, 2, 2, 1, 0).

The minimum vertex degree in a graph G is denoted , and the maximum degree is denoted .

A graph whose degree sequence contains only multiple copies of a single integer is called a regular

graph.

It is possible for two topologically distinct graphs to have the same degree sequence.

Chapter-IV

Graph Theory

End Chapter quizzes :IV

Ques 1. A graph which starts and ends on the same vertex is called a

a. Line

b. Circle

c. loop

d. Vertices

Ques 2. A graph with neither loops nor parallel edges is called

a. multigraph

b. simple graph

c. pseudograph

d. Complete graph

Ques 3. A complete graph with n nodes represents the edges of an

a. (n-1)-simple

b. (n)- simplex

c. (n+1)- simplex

d. 0

Ques 4. A triangle is a

a. bipartite graph

b. non-bipartite graph

c. Loop

d. None

Ques 5.All trees are bipartite

a. Always True

b. Always False

c. Not defined

d. None

Ques 6.The degree (or valency) of a vertex of a graph is the number of edges incident to the vertex,

with loops counted.

a. Once

b. Twice

c. Thrice

d. Four times

Ques 7.The complete graph with n graph vertices is denoted Kn and with undirected edges

a. n

b. n (n-1)!

c.

d. n (n +1)!

Ques 8 For any graph the minimum degree could be

a. 3

b. 2

c. 1

d. 0

Ques 9. A trivalent graph is one that is regular of degree

a. 3

b. 2

c. 1

d. 0

Ques 10.

a. simple graph with four vertices

b. non simple graph with four vertices

c. simple graph with three vertices

d. non simple graph with three vertices

Chapter-V

Data Analysis

Contents:

5.1 Data and Statistical data

5.2 Frequency Distribution

5.3 Graphical presentation of Frequency distribution

5.4 Measure of Central tendency

5.5 Measure of Dispersion

Data Analysis

5.1 Data and Statistical data:

Statistics is a branch of applied mathematics concerned with the collection and interpretation of

quantitative data and the use of probability theory to estimate population parametersStatistical

methods can be used to summarize or describe a collection of data; this is called descriptive

statistics.

Data: A collection of values to be used for statistical analysis.

A dictionary defines data as facts or figures from which conclusions may be drawn. Data

may consist of numbers, words, or images, particularly as measurements or observations of a set of

variables. Data are often viewed as a lowest level of abstraction from which information and

knowledge are derived. Thus, technically, it is a collective or plural noun.

Datum is the singular form of the noun data. Data can be classified as either numeric or

nonnumeric. Specific terms are used as follows:

Types of Data

I. I Qualitative data are nonnumeric.

1. {Poor, Fair, Good, Better, Best}, colors (ignoring any physical causes), and types of

material {straw, sticks, bricks} are examples of qualitative data.

2. Qualitative data are often termed categorical data. Some books use the terms individual

and variable to reference the objects and characteristics described by a set of data. They

also stress the importance of exact definitions of these variables, including what units they

are recorded in. The reason the data were collected is also important.

II Quantitative data are numeric.

Quantitative data are further classified as either discrete or continuous.

Discrete data are numeric data that have a finite number of possible values.

o A classic example of discrete data is a finite subset of the counting numbers,

{1,2,3,4,5} perhaps corresponding to {Strongly Disagree... Strongly Agree}.

When data represent counts, they are discrete. An example might be how many

students were absent on a given day. Counts are usually considered exact and integer.

Continuous data have infinite possibilities: 1.4, 1.41, 1.414, 1.4142, 1.141421...

The real numbers are continuous with no gaps or interruptions. Physically

measureable quantities of length, volume, time, mass, etc. are generally considered

continuous. At the physical level (microscopically), especially for mass, this may

not be true, but for normal life situations is a valid assumption.

Data analysis is a process of gathering, modeling, and transforming data with the goal of

highlighting useful information, suggesting conclusions, and supporting decision making. Data

analysis has multiple facets and approaches, encompassing diverse techniques under a variety of

names, in different business, science, and social science domains.

5.2 Frequency Distribution:

The distribution of empirical data is called a frequency distribution and consists of a count of the

number of occurrences of each value. If the data are continuous, then a grouped frequency

distribution

is used. Typically, a distribution is portrayed using a frequency polygon or a histogram.

Mathematical distributions are often used to define distributions. The normal distribution is,

perhaps, the best known example. Many empirical distributions are approximated well by

mathematical distributions such as the normal distribution.

Grouped Frequency Distribution A grouped frequency distribution is a frequency distribution in

which frequencies are displayed for ranges of data rather than for individual values. For example,

the distribution of heights might be calculated by defining one-inch ranges. The frequency of

individuals with various heights rounded off to the nearest inch would be then be tabulated.

5.3 Graphical presentation of Frequency distribution:

Histogram: A histogram is a graphical display of tabulated frequencies. A histogram is the graphical version of

a table that shows what proportion of cases fall into each of several or many specified categories.

Example of a histogram of 100 values

Advantages

Visually strong

Can compare to normal curve

Usually vertical axis is a frequency count of items falling into each category

Disadvantages

Cannot read exact values because data is grouped into categories

More difficult to compare two data sets

Use only with continuous data

Frequency Polygons:

Frequency polygons are a graphical device for understanding the shapes of distributions. They

serve the same purpose as histograms, but are especially helpful in comparing sets of data.

Frequency polygons are also a good choice for displaying cumulative frequency distributions.

To create a frequency polygon, start just as for histograms, by choosing a class interval. Then draw

an X-axis representing the values of the scores in your data. Mark the middle of each class interval

with a tick mark, and label it with the middle value represented by the class. Draw the Y-axis to

indicate the frequency of each class. Place a point in the middle of each class interval at the height

corresponding to its frequency. Finally, connect the points. You should include one class interval

below the lowest value in your data and one above the highest value. The graph will then touch the

X-axis on both sides.

Advantages

Visually appealing

Can compare to normal curve

Can compare two data sets

Disadvantages

Anchors at both ends may imply zero as data points

Use only with continuous data

Frequency Curve:

A smooth curve which corresponds to the limiting case of a histogram computed for a frequency distribution

of a continuous distribution as the number of data points becomes very large.

Advantages

Visually appealing

Disadvantages

Anchors at both ends may imply zero as data points

Use only with continuous data

5.4 Measure of Central tendency:

Central Tendency is the center or middle of a distribution. There are many measures of central

tendency. The most common are the mean, median and mode.

The center of a distribution could be defined three ways:

1. the point on which a distribution would balance,

2. the value whose average absolute deviation from all the other values is minimized, and

3. the value whose squared difference from all the other values is minimized.

From the simulation in this chapter, you discovered (we hope) that the mean is the point on which

a distribution would balance, the median is the value that minimizes the sum of absolute

deviations, and the mean is the value that minimizes the sum of the squared values.

Arithmetic Mean:

The arithmetic mean is the most common measure of central tendency. For a data set, the mean is

the sum of the observations divided by the number of observations. Basically, the mean describes

the central location of the data.

For a given set of data, where the observations are x1, x2,….,xi ; the Arithmetic Mean is defined

as :

The weighted arithmetic mean is used, if one wants to combine average values from samples of

the same population with different sample sizes:

Example 1:

Observations 12 15 20 22 30

Weights 2 5 7 6 1

Find the mean.

Observations Weights xiwi

Mean =401

/21 =19.10

12 2 24

15 5 75

20 7 140

22 6 132

30 1 30

Total 21 404

Mean

Advantages

can be specified using and equation, and therefore can be manipulated algebraically

is the most sufficient of the three estimators

is the most efficient of the three estimators

is unbiased

Disadvantages

is very sensitive to extreme scores (i.e., low resistance)

value is unlikely to be one of the actual data points

requires an interval scale

anything else about the distribution that we’d want to convey to someone if we were

describing it to them?

Median:

The median of a finite list of numbers can be found by arranging all the observations from lowest

value to highest value and picking the middle one. If there is an even number of observations, the

median is not unique, so one often takes the mean of the two middle values.

For Odd number of observations:

Median = (n+1)/2 th

observations.

For Even number of observations:

Median = Average of (n/2) th

and (n/2 + 1) th

observations.

Here are the sample test scores you have seen so often:

100, 100, 99, 98, 92, 91, 91, 90, 88, 87, 87, 85, 85, 85, 80, 79, 76, 72, 67, 66, 45

The "middle" score of this group could easily be seen as 87. Why? Exactly half of the scores lie

above 87 and half lie below it. Thus, 87 is in the middle of this set of scores. This score is known

as the median.

In this example, there are 21 scores. The eleventh score in the ordered set is the median score (87),

because ten scores are on either side of it.

If there were an even number of scores, say 20, the median would fall halfway between the tenth

and eleventh scores in the ordered set. We would find it by adding the two scores (the tenth and

eleventh scores) together and dividing by two.

Median

Advantages

is unbiased

is unaffected by extreme scores (i.e., high resistance)

doesn’t require the use of an interval scale, as long as you can order the scores along

some continuum then you can find the median

Disadvantage

can not be specified using an equation so can’t be manipulated algebraically

is the least sufficient of the three estimators

is less efficient than the mean

Mode:

The mode is the most frequently occurring value. It is the most common value in a distribution:

The mode of 3, 4, 4, 5, 5, 5, 8 is 5. Note that the mode may be very different from the mean

and the median.

With continuous data such as response time measured to many decimals, the frequency of each

value is one since no two scores will be exactly the same. Therefore the mode of continuous

data is normally computed from a grouped frequency distribution. The grouped frequency

distribution table shows a grouped frequency distribution for the target response time data.

Since the interval with the highest frequency is 600-700, the mode is the middle of that interval

(650).

Range Frequency

500-600 3

600-700 6

700-800 5

800-900 5

900-1000 0

1000-1100 1

Table 3: Grouped frequency

distribution

Mode

Advantages

represents a number that actually occurred in the data

represents the largest number of scores, and so the probability of getting that score is

greater then the probability of getting any of the other scores if an observation is just

chosen at random is unaffected by extreme scores (i.e., high resistance)

is unbiased

doesn’t require an interval scale

Disadvantages

the mode depends on how we group the data

can not be specified using an equation so can’t be manipulated algebraically

is less sufficient than the mean

is less efficient than the mean

5.5 Measure of Dispersion:

Measures of Dispersion provide us with a summary of how much the points in our data set vary,

e.g. how spread out they are or how volatile they are.

In measuring dispersion, it is necessary to know the amount of variation and the degree of

variation. The former is designated as absolute measures if dispersion and expressed in the

denomination of original variants while the latter is designated as related measures of dispersion.

Absolute measures can be divided into positional measures based on some items of the

series such as (I) Range, (ii) Quartile deviation or semi – interquartile range and those which are

based on all items in series such as (I) Mean deviation, (ii) Standard deviation. The relative

measures in each of the above cases are called the coefficients of the respective measures. For

purposes of comparison between two or more series with varying size or number of items, varying

central values or units of calculation, only relatives measures can be used.

The following are the important methods of studying variation :

1. Range

2. Mean deviation

3. Standard deviation and Variance (which is closely related to standard deviation)

4. The Coefficient of Variation

Range :Range is the simplest of the summary measures of variation .It is also the crudest and most

prone to error .It is computed as the difference between the largest and the smallest value in a data

set:

Range = H- L

Absolute range H-L Relative range; Coefficient of range = ——————————— = ———

Sum of the two extremes H+L

For example, for the data set {2, 2, 3, 4, 14}

Range = 14-2=12 14 – 2 12

Coefficient of range = ———— = —— = 0.75

14 + 2 16

Example :

You are given the following data:

Compute the sample range

Solution :

H = 11, L = 3

range = H - L = 11 - 3 = 8

MeanDeviation: Mean Deviation can be calculated from any value of Central Tendency, viz. Mean, Median, Mode.

Accordingly, Mean Deviation can be of the following types:

Mean Deviation about Mean

Mean Deviation about Median

Mean Deviation about Mode

Mean Deviation about Mean =

Properties of Mean Deviation about Mean :-

The average absolute deviation from the mean is less than or equal to the Standard Deviation.

The mean deviation of any data set from its mean is always zero.

The mean absolute deviation is the average absolute deviation from the mean and is a common

measure of Forecast Error or Time Series Analysis.

For example, for the data set {2, 2, 3, 4, 14}:

Measure of central

tendency Absolute deviation

Mean = 5

Variance and standard deviation:

Variance and standard deviation are the most common of all of the measures of variation

Variance is a measure of statistical dispersion, indicating how its possible values are spread around

the mean. Thus, variance indicates the variability of the values. A smaller value implies a smaller

variation from the mean

The positive square root of Variance is called the Standard Deviation.

Let us consider an example:

Values Xi - Mean(x) [Xi - Mean(x)]2

4 -1 1

6 1 1

5 0 0

5 0 0

Total =20 , mean=5 2

Variance = ¼ .2 =1/2

S.D =

The Coefficient of Variation:

The Coefficient of Variance is a measure of variation expressed as a percentage the sample mean:

Chapter-V

Data Analysis

End Chapter quizzes :

Ques 1. Singular form of the data is a. Datum b. Stratum c. Date d. Data

Ques 2. Graphical presentation of Frequency distribution can be done by

a. Histogram

b. Frequency polygons

c. Frequency Curve

d. All the three

Ques 3. Which one is unaffected by extreme scores

a. Mean

b. Median

c. Mode

d. Range

Ques 4.Which one is not the Measures of Dispersion

a. Range

b. Mean deviation

c. Histogram

d. Standard deviation

Ques 5.Chaya took 7 math tests in one marking period. What is the range of her test scores?

89, 73, 84, 91, 87, 77, 94

a. 25

b. 21

c. 13

d. 15

Ques 6.In a crash test, 11 cars were tested to determine what impact speed was required

to obtain minimal bumper damage. Find the mode of the speeds given in miles per hour

below.

24, 15, 18, 20, 18, 22, 20, 26, 18, 26, 24

a. 18

b. 20

c. 18.6

d. 15

Ques 7. A survey conducted by an automobile company showed the number of cars per household

and the corresponding probabilities. Find the standard deviation.

Number of cars X 1 2 3 4

Probability P(X) 0.32 0.51 0.12 0.05

a. 4.24

b. 0.63

c. 0.79

d. 1.9

Ques 8. The given data shows the number of burgers sold at a bakery in the last 14 weeks.

17, 13, 18, 17, 13, 16, 18, 19, 17, 13, 16, 18, 20, 19

Find the median number of burgers sold.

a. 18.5

b. 17

c. 18

d. 17.5

Ques 9.Histograms can be constructed for

a. Discrete data

b. Continuous data

c. Both

d. none

Ques 10.Which is called positional average

a. Mean

b. Median

c. Mode

d. None

Key to End Chapter Quizzes.

Chapter-I

SET THEORY

Key

1. c

2. c

3. a

4. d

5. a

6. b

7. a

8. d

9. a

10. a

1 (b); 2(d); ……………………………………………10(c)

Chapter-II

MATHEMATICAL LOGIC

1. b and c 2. c 3. a 4. d 5. a 6. b 7. a 8. c 9. c 10. a

Chapter-III

MODERN ALGEBRA

Key

1. a

2. b

3. a

4. a

5. c

6. c

7. b

8. d

9. d

10. b

Chapter-IV

Graph Theory

Key

1. c

2. b

3. a

4. b

5. a

6. b

7. c

8. d

9. a

10. b

Chapter-V

Data Analysis

Key

1. a

2. d

3. b

4. c

5. b

6. a

7. c

8. b

9. b

10. b