Download Discrete mathematics gv kumbhojkar and more Schemes and Mind Maps Discrete Mathematics in PDF only on Docsity!
REVISED SYLLABUS
S.Y.B.SC. (I.T.), SEM. - III,
LOGIC AND DISCRETE MATHEMATICS
Unit 1 :
Set Theory : Fundamentals - Sets and subsets, Venn
Diagrams, Operations on sets, Laws of Set Theory, Power
Sets and Products, Partition of sets, The Principle of Inclusion
Logic : Propositions and Logical operations, Truth tables,
Equivalence, Implications, Laws of Logic, Normal forms,
Predicates and quantifiers, Mathematical Induction.
Unit 2 :
Relations, diagraphs and lattices : Product sets and
partitions, relations and diagraphs, paths in relations and
diagraphs, properties of relations, equivalence and partially
ordered relations, computer representation of relations and
diagraphs, manipulation of relations, Transitive closure and
Warshall’s algorithm, Posets and Hasse Diagrams, Lattice.
Unit 3 :
Functions and Pigeon Hole Principle : Definitions and types
of functions : injective, subjective and bijective, Composition,
identity and inverse, Pigeon hole principle.
Unit 4 :
Graphs and Trees : Graphs, Euler paths and circuits,
Hamiltonian paths and circuits, Planner graphs, coloring
graphs, Isomorphism of Graphs.
Trees : Trees, rooted trees and path length in rooted trees,
Spanning tree and Minimal Spanning tree, Isomorphism of
trees, Weighted trees and Prefix Codes.
Unit 5 :
Algebric Structures : Algebraic structures with one binary
operation - semi groups, monoids and groups, Product and
quotient of algebraic structures, Isomorphism, homomorphism,
automorphism, Cyclic groups, Normal sub group, codes and
group codes, Algebraic structures with two binary operations -
rings, integral domains and fields. Ring homomorphism and
Isomorphism.
Unit 6 :
Generating Functions and Recurrence relations : Series
and Sequences, Generating Functions, Recurrence relations,
Applications, Solving difference equations, Fibonacci.
Books :
Discrete mathematical structures by B Kolman RC Busby, S
Ross PHI Pvt. Ltd.
Discrete mathematical structures by R M Somasundaram
(PHI) EEE Edition.
Reference :
Discrete structures by Liu, Tata McGraw - Hill.
Digital Logic John M Yarbrough Brooks / cole, Thompson
Learning
Discrete Mathematics and its Applications, Kenneth H. Rosen,
Tata McGraw - Hill.
Discrete Mathematics for computer scientists and
Mathematicians, Joe L. Mott, Abraham
Kandel Theodore P. Baker, Prentice - Hall of India Pvt. Ltd.
Discrete Mathematics With Applications, Susanna S. Epp,
Books / Cole Publishing Company.
Discrete Mathematics, Schaum’s Outlines Series, Seymour
Lipschutz, Marc Lipson, Tata McGraw - Hill.
1.2 SETS AND SUBSETS
1.2.1 Sets
A set is any well defined collection of distinct objects.
Objects could be fans in a class room, numbers, books etc.
For example, collection of fans in a class room collection of
all people in a state etc. Now, consider the example, collection of
brave people in a class. Is it a set? The answer is no because brave
is a relative word and it varies from person to person so it is not a
set.
Note : Well-defined means that it is possible to decide whether a
given object belongs to given collection or not.
Objects of a set are called as elements of the set. Sets are
denoted by capital letters such as A, B, C etc and elements are
denoted by small letters x, y, z etc.
If x is an element of set A then we write x Aand if x is not
an element of A then we write x A.
There are two ways to represent a set one way by listing all
the elements of a set separated by a comma enclosed in braces.
Another way of specifying the elements of a set is to give a rule for
set membership.
For example, A = (^) e, t, a (^) can be written as
A = x x | is a letter in the word 'eat'
We have following Basic sets of numbers.
(a) = set of all Natural numbers.
= (^) 1, 2, 3,......
(b) = set of all whole numbers
= (^) 0, 1, 2, 3, ......
(c) = set of all Integers
= (^) ...., – 2, –1, 0,1, 2, 3,......
(d) (^) = set of all rational numbers.
p
/p, q q 0
q
,
(e) = set of all real numbers.
1.2.2 Some Basic Definitions –
(a) Empty Set : A set without any element. It is denoted by or { }
For examples,
B = (^) x x | 1 and x =
C = (^) x x | and x +1 = 1 =
(b) Equal Sets :- Two sets A and B are said to be equal if they have
same elements and we write A = B.
For examples,
(1) A = (^) x x | is a letter in the word 'ate'
B = (^) y | y is a letter in the word 'eat'
A = B
(2) (^)
2
X = –3, 3 and Y = x x | = 9, x
i.e. X = Y
(c) Subset :- Set A is said to be a subset of B if every element of A
is an element of B and this is denoted by A B or B A. If A
is not a subset of B we write A B.
For example,
(1) (^)
A = 1 , B = | = 1
2
x x , x then^ A^ ^ B and B^ A
Note : (1) Every set A is a subset of itself i.e. A A
(2) If A Bbut A Bthen we say A is a proper subset
of B and we write A B. If A is not a proper subset
of B then we write A B.
(3) A for any set ‘A’
(4) A = B iff A Band B A
(d) Finite Set :- A set A with ‘n’ distinct elements, (^) n (^) is called
as a finite set.
For example,
(1) A = (^) x | x , 5 x 2 0
(2) B = (^) y | y is a hair on some ones head
For example,
(1) A = {a, b, d} (2) B = {5, 6, 7}
A
a
b
d
B
Fig. 1.1 Fig. 1.
Check your progress :
- Identify each of the following as true or false.
(a) A = A (b) A A (c) A A (d) A
(e) A (f) If A {1} then P(A) = { , A}
2. If A = { x , y, 3 }, then find (a) P(A), (b) |A| (c) |P(A)|
- Which of the following are empty sets?
(a) (^) x x | ,1 x (^2)
(b)
| , = –
2
x x x
(c) (^) x x | , x +1 = 1
(d)
| , = 3
2
x x x
- Draw the Venn diagram for .
1.3 OPERATION ON SETS
1.3.1 Basic definitions :
(a) Union of two sets :- Let A and B be two given sets. Union of A
and B is the set consisting of all elements that belong to ‘A’ or
‘B’ and it is denoted by A B. A B = x x | A or x B
For example,
(1) A = (^) x , y, z , B = (^) 2, 5
A B = (^) x , y, z, 2, 5
(2) A = , B =
A B = = 1, 2, 3, ..... ..... , – 2, –1, 0,1, 2, ....
= (^) ....., – 2, –1, 0,1, 2, .... =
Note : (1) If A Bthen A B = B
(2) A A = A
(3) A = A
A
B
Fig. 1.
(b) Intersection of two sets :- Let A and B be two given sets.
Intersection of A and B is the set consisting of the elements
Present in A and B. (i.e. in both) and it is denoted by A B.
A B = (^) x x | A and x B
For example,
(1) A = {1, 2, 3}, B = {2, 4, 5} then A B = (^) 2
(2) A = {1, 2, 3} and B = { x , y } then A B = such sets whose
intersection is empty is called as disjoint sets.
Note : (1) A A = A
(2) If A Bthen A B = A
(3) A =
(4) Shaded region represents A B (5) Disjoint sets
A (^) B
U
A B
U
Fig. 1.4 Fig. 1.
Definition for union and Intersection can be extended to ‘n’
number of sets. ( n )
(c) Complement of a set :- Let U be a given universal set and let A
be any subset of U. Then complement of a set A in U is set of
those elements which are present in U but not in A and it is
denoted by
c
A or A’ or A.
i.e. (^)
c
A = x x | A & x U
Shaded region represents A B. (4)
Check your progress :
- If A = U ( U – universal set), then (a)
c
A , (b) A \ U, (c) U \ A,
(d) A U
- If U = (^) x x | and x (^17) and A = {1, 3, 5, 6}, B = {3, 4, 7,
5, 8}, then (a)
c
A , (b) A\B, (c) B\A, (d) A B, (e)
c
B , (f) A B,
(g) A B.
1.3.2 Algebraic Properties of set operations
Like Algebraic properties of Real numbers, sets also satisfy
some Algebraic Properties with respect to the operations union,
intersection etc.
(I) Commutative Properties
(1) A B = B A
(2) A B = B A
(II)Associative Properties
(3) A (^) B C = (^) A B (^) C
(4) A (^) B C (^) = A B C
(III) Distributive Properties
(5) A (^) B C (^) = (^) A B (^) (^) A C
(6) A (^) B C (^) = (^) A B (^) (^) A C
(IV) Idempotent Properties
(7) A A = A
(8) A A = A
(V) Properties of Complement
A = A
(10) A A = U
(11) A A =
(12) = U
(13) U =
(14) A B =A B
(15) A B =A B
(De Morgan’s laws)
Properties (1) to (13) can be proved easily. We will prove (14) and
(15) here.
(14) A B = A B
Proof : (^)
A B = x x | A B and x U
= (^) x | (^) x A and x U (^) and (^) x B and x U
x x | A and x B
= A B
Similarly, we can prove (15).
Example 1: Prove that i) ( ) ( )
c
A B A B A and
ii) ( ) ( )
c
A B A B A.
Solution: L.H.S.= ( ) ( )
c
A B A B
c
A B B ( Distributive law)
= A (
c
B B complement law)
= A
= R.H.S
Hence ( ) ( )
c
A B A B A.
Similarly, we can prove ( ) ( )
c
A B A B A.
Example 2: If U = {x\x is a natural number less than 20} is the
universal
set, A = {1, 3, 4, 5, 9}, B = {3, 5, 7, 9, 12}. Verify that
De Morgan’s laws.
Solution: De Morgan’s laws can be state as i) A B = A B,
ii) (^) A B = A B.
By listing method,
U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19},
and A = {1, 3, 4, 5, 9},
A = {2, 6, 7, 8, 10, 11, 12, 13,14,15,16,17,18,19},
and B = {3,5,7,9,12},
B = {1, 2, 4, 6, 8, 10, 11, 13, 14, 15, 16, 17, 18,19}
A B = {1, 3, 4, 5, 7, 9, 12}
( A B )= {2, 6, 8, 10, 11, 13, 14, 15, 16, 17, 18, 19}
Also ( A B )= {2, 6, 8, 10, 11, 13, 14, 15, 16, 17, 18, 19}
Theorem :- If A, B and C be given finite sets, then
A B C = A + B + C – A B – B C – A C + A B C
Example 5
In a survey of people it was found that 80 people knew
Maths, 60 knew physics, 50 knew chemistry, 30 new Maths and
Physics, 20 knew Physics and Chemistry, 15 knew Maths and
Chemistry and 10 knew all three subjects. How many people knew?
(a) At least one subject
(b) Maths only
(c) Physics only
(d) Maths and Chemistry only
Solution : Let M, P, C represents respectively, the set of students
knowing Maths, Physics and Chemistry.
|M| = 80, |P| = 60, |C| = 50, M P = 30, M C = 15,
P C = 20, M P C = 10
(a) By addition principle
M P C = M + P + C – M P – M C – P C + M P C
Let’s draw the Venn diagram of above situation.
5
20 20
5
10
10
25
M P
C
Fig. 1.
(b) Maths only = 80 – (20 + 10 + 5)
(c) Physics only = 60 – (20 + 10 + 10) = 20
(d) Maths and Chemistry only = 15 – 10 = 5
Example 6: Out of 150 residents a building, 105 speak Marathi, 75
speak Gujarati and 45 speak both Languages. Find the number of
residents who do not speak either of the languages also find the
number of residents who speak only Marathi.
Solution:- Let A be the set of resident who speak Marathi.
B the set of resident who speak Gujarati
Given │U│=
│A│=
│B│=
│A∩B│=
By principal of Inclusion-Exclusion.
│AUB│=│A│+│B│-│A∩B│
i). Number of resident who do not speak either of language.
│(AUB)’│=│U│-│AUB│
ii). The number of resident who speak only Marathi
=│A│-│A∩B│
Example 7: Out of 240 students in college 130 students are in
N.C.C. 110 are in N.S.S. and 80 are in other activity in this 40 are
N.C.C. and N.S.S both, 35 are N.C.C and other activity and 30 are
N.S.S. and other activity but 20 student are take part in all three.
Find the number of students takes part in
i). Atleast any one.
ii). None of them.
iii). Only N.S.S.
iv). Only other activity.
v). Only N.S.S and N.S.S but not in other activity.
Solution:- Let A be the set of N.S.S students.
B be the set of N.C.C students.
C be the set of other activity student.
Here │A│=130, │B│=110, │C│=80.
│A∩B│=40, │A∩C│=30, │B∩C│=35, │A∩B∩C│=20.
By principle Inclusion-Exclusion
i). atleast one of them i.e.│AUBUC│
│AUBUC│=│A│+│B│+│C│-│A∩B│-│B∩C│-
│A∩C│+│A∩B∩C│
i) AUBUC A B C A B B C A C A B C
ii). 2 or 3 but not by 5.
=│AUB│-│A∩B∩C│
iii). Only by 5.
=│C│-│A∩C│-│B∩C│+│A∩B∩C│
1.7 LET US SUM UP
This chapter consist of sets and different operations on sets
with different examples which helps in better understanding of the
concept and able to use in different areas. We saw the principle of
inclusion – Exclusion which can be used in different counting
problems we saw some concepts of number theory such as division
in Integers, sequence etc. which is useful in computer security. At
the end we saw definition of a mathematical structure and it is
different properties.
1.8 REFERENCES FOR FURTHER READING
a) Discrete structures by Liu.
b) Discrete mathematics its Application, Keneth H. Rosen TMG.
c) Discrete structures by B. Kolman HC Busby, S Ross PHI Pvt.
Ltd.
d) Discrete mathematics, schaum’s outlines series, seymour Lip
Schutz, Marc Lipson, TMG.
1.9 UNIT END EXERCISES
- Let
2
A = x x | and x + 7 = 0 , B = (^) x x | ,
C = (^) x x | , 0 < x <0.2, D (^) x x | 6 , q q
E = (^) x x | , x + 7 = 7
Check whether following are True or False.
(i) A is finite, (ii) B A, (iii) (^) E = A , (iv) E A D,
(v) C is infinite, (vi) B = , (vii) A E, (viii) B C = A
- Prove A – B = A – (A B)
- There are 250 students in a computer Institute of these 180
have taken a course in Pascal, 150 have taken a course in
C++, 120 have taken a course in Java. Further 80 have taken
Pascal and C++, 60 have taken C++ and Java, 40 have taken
Pascal and Java and 35 have taken all 3 courses. So find –
(a) How many students have not taken any course?
(b) How many study atleast one of the languages?
(c) How many students study only Java?
(d) How many students study Pascal and C++ but not Java?
(e) How many study only C++ and Java?
- The students stay in hostel were asked whether they had a
textbook r a digest in their rooms. The results showed that
650 students had a textbook, 150 deed not have a textbook,
175 had a digest and 50 had neither a textbook nor a digest.
Find, i). the number of students in hostel , ii).How many have
a textbook and digest both, iii). How many have only a digest.
- Prove that (B
c
∩U)∩(A
c
U )=(AUB)
c
.
- Prove that , i).AU(A∩B)=A, ii). A∩(AUB)=A.
- In a survey of 80 people in Gokuldham 50 of them drink Tea,
40 of them drink Coffee and 20 drink both tea and coffee.
Find the number of people who take atleast one of the two
drinks also find the number of students who do not take tea or
Coffee.
- In a survey of 60 people, It was found that 25 read magazine.
26 read Times of India and 26 read DNA. Also 9 read both
magazine and DNA, 11 read both magazine and times of
India , 8 read times of India and DNA and 8 are not reading
anything.
i). Find the number of people who read all three.
ii). Draw a Vann diagram.
iii). Determine the number of people who read exactly one
magazine.
- It will rain today.....
- Mumbai is capital city of Maharashtra.
- Do you know where is Vijay?
4. 2 × 3 − 5 = 1.
2
x − 1 = 4.
- Come in!
In above, sentences (1), (2) and (4) are statements. (3) is not a
statement as it is question,(5) is declarative but depending upon the
value of x it is true or false. Sentence (6) is a command and hence
not a statement.
2.1.1 Logical connectives and compound statements.
Just as in mathematics variables x, y, z, ... can take real values
and can be combined by operations +,−,×,÷, in logic, the variables p,
q, r, .. can the replaced by statements. The variable p, q, r, .... are
called as propositional variables. For example we can write p : Sonia
Gandhi is president of India, q : Newton was a Physicist, r : It will
rain today. etc. One can combine propositional variables by logical
connectives to obtain more complex statements - compound
statement. For example suppose Q Mangoes are ripe, R : Oranges
are sour. The statement Q and R means Mangoes are ripe and orange
is sour. The truth value of compound statement depends on truth
values of statements which are combined and on the logical
connectives that are used. In this subsection, we will discuss most
commonly used logical connectives.
2.1.2 Negation :
Suppose P is any statement. Then negation of P, denoted
by p. Thus if P is true then p is false and vice a versa. A table
giving truth values of compound statement in terms of compound
parts is known as truth table.
p
p
T
F
F
T
Strictly speaking not P is not compound statement as it is unary
operation.
Example 1 Give negation of
- p: It is hot.
- q: 2 is a divisor of 5.
Solution :
- ~ P : It is not the case that it is hot i.e. it is not hot.
- ~ q : 2 is not divisor of 5. Since q is false, (^) q is true.
2.1.3 Conjunction
The next operation is conjunction. If p and q are two
statement then conjunction of p and q is the compound statement “p
and q”. The notation is p q. The operation and is a binary
operation on the set of statements. The (^) p q is true whenever both p
and q on true, false otherwise. Thus the truth table is given by
p q
p q
T
T
F
F
T
F
T
F
T
F
F
F
Example 2 Form the conjunction of p and q.
- p: I will drive my car q: I will reach the office in time.
- p: 2 is even q: 11 is odd.
- p: 2 + 3 + 1 = 6 q: 2 + 3 > 4
- p: Delhi is capital of India q: Physics is a science subject.
Solution :
- (^) p q is “I will drive my car and I will reach office in time”.
- “2 is even and 11 is odd”.
- “2 + 3 + 1 = 6 and 2 + 3 > 4”.
- “Delhi is capital of India and Physics a science subject”.
2.1.4 Disjunction
The second logical connective used is disjunction.
Disjunction of statements p and q is dented by p q ,which means p
or q. The statement p q is true where p or q or both are true and
is false only when both p and q are false.
The truth table for p q is as follows.
p q
p q
T
T
F
F
T
F
T
F
T
T
T
F
