Chapter 12.pptx

Lecture delivered for training the trainers on How to teach Discrete
Mathematics at SCERT, Chennai
Jacob(II) Bernoulli
Born: 17 October 1759 in Basel,
Switzerland
Died: 3rd July 1789 in St
Petersburg, Russia

Discrete Objects
• Discrete objects can often be enumerated by
integers
• Examples of discrete objects include buildings,
roads, and parcels.
• Discrete objects are usually nouns.

Discrete Mathematics
• Discrete mathematics is the study of mathematical
structures that are fundamentally discrete rather than
continuous. ...
• Discrete mathematics therefore excludes topics in
"continuous mathematics" such as calculus and
analysis.
• Some of the branches of Discrete Mathematics are
combinatorics, mathematical logic, boolean algebra,
graph theory, coding theory etc.

Unary operation
𝜓: 𝐴 ⟶ 𝐴
𝑇ℎ𝑎𝑡𝑖𝑠∀𝑎, ∈ 𝐴, 𝜓(𝑎) ∈ 𝐴
That is , any function from a non -empty set into itself
is an unary operation .
𝐿𝑒𝑡𝐴𝑏𝑒ℙ(𝑆) − 𝑡ℎ𝑒𝑝𝑜𝑤𝑒𝑟𝑠𝑒𝑡𝑜𝑓𝑆
Then for a subset C of A if we associate the
complement of C in A is an unary operation .
𝐿𝑒𝑡𝐴𝑏𝑒𝕄𝑛×𝑛
Then for a matrix C if we associate the adjoint of
C in is an unary operation .

Binary operations
∗: 𝐴 × 𝐴 ⟶ 𝐴
𝑇ℎ𝑎𝑡𝑖𝑠∀𝑎, 𝑏 ∈ 𝐴, 𝑎 ∗ 𝑏 ∈ 𝐴
⟨𝐴,∗⟩𝑖𝑠𝑐𝑎𝑙𝑙𝑒𝑑𝑎𝑛𝑎𝑙𝑔𝑒𝑏𝑟𝑎𝑖𝑐𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒
Examples
1)⟨𝑁, +⟩𝑖𝑠𝑐𝑎𝑙𝑙𝑒𝑑𝑎𝑛𝑎𝑙𝑔𝑒𝑏𝑟𝑎𝑖𝑐𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒
2)⟨𝑁,×⟩𝑖𝑠𝑐𝑎𝑙𝑙𝑒𝑑𝑎𝑛𝑎𝑙𝑔𝑒𝑏𝑟𝑎𝑖𝑐𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒
3)⟨𝑁, −⟩𝑖𝑠𝑐𝑎𝑙𝑙𝑒𝑑𝑛𝑜𝑡𝑎𝑛𝑎𝑙𝑔𝑒𝑏𝑟𝑎𝑖𝑐𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒
4)⟨𝑁,/⟩𝑖𝑠𝑐𝑎𝑙𝑙𝑒𝑑𝑛𝑜𝑡𝑎𝑛𝑎𝑙𝑔𝑒𝑏𝑟𝑎𝑖𝑐𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒
Let N be the set of all Natural numbers

Algebraic structure
Examples
⟨𝐴,∗⟩𝑖𝑠𝑎𝑛𝑎𝑙𝑔𝑒𝑏𝑟𝑎𝑖𝑐𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒
Let A = Z , Q , R be respectively the set of all integers,
rational , real, complex numbers numbers, then
Check for which arithmetic operation
On the set of all integers verify the operation is binary
⟨𝑍,∗⟩𝑔𝑖𝑣𝑒𝑛𝑏𝑦𝑚 ∗ 𝑛 = 𝑚 + 𝑛 − 𝑚. 𝑛
𝑌𝐸𝑆

Properties of Binary Operation 𝐿𝑒𝑡⟨𝐴,∗⟩𝑏𝑒𝑎𝑛𝑎𝑙𝑔𝑒𝑏𝑟𝑎𝑖𝑐𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒
Sl.No Property If
1 Commutative
2 Associative
3
Existence of
Identity
4
Existence of
Inverse
Identity and inverse if exists then they are unique w.r.to
the binary operation

on and multiplication satisfies all the properties mentioned earlier.
The additive identity is ‘0 ‘
Inverse for any ’n’ under addition is ‘ - n’
The multiplicative identity is ‘1’
Inverse for any non -zero number ‘x’ is 1/x
𝑎 ∗ 𝑏 = 𝑎 𝑏𝑖𝑠𝑛𝑜𝑡𝑎𝑏𝑖𝑛𝑎𝑟𝑦𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑖𝑛ℝ
𝐵𝑒𝑐𝑎𝑢𝑠𝑒𝑎 ∗ −1 = 𝑎 −1 ∉ ℝ

Is the addition and multiplication, on the set of all
prime numbers , a binary operation ?
Is the addition, on the set of all Fibonacci numbers ,
a binary operation ?
NO because 2 and 13 are Fibonacci numbers but 2+13
= 15 is not so.
NO because

Boolean Matrices
A Boolean matrix is of the form

• Remark: The operations of Join & Meet defines a binary
operations on the the set of all Boolean matrices .
• Those two binary operations are commutative and
associative which posses identity but no inverse

Operation table for a finite set with binary operation
Each row is a permutation
Binary operation is non -
commutative
Is the Binary operation
associative
c a
a
Fill in the following table so that
the binary operation ∗ on
A = {a, b, c} is commutative

Modulo operations
𝑂𝑛𝑡ℎ𝑒𝑠𝑒𝑡𝑜𝑓𝑎𝑙𝑙𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠ℤ
𝐷𝑒𝑓𝑖𝑛𝑒𝑓𝑜𝑟𝑚, 𝑛 ∈ ℤ𝑚 +𝑝 𝑛𝑎𝑠,
the remainder when m+n is divided by p
𝑇ℎ𝑒𝑛 +𝑝 𝑖𝑠𝑎𝑏𝑖𝑛𝑎𝑟𝑦𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑜𝑛ℤ
⟨ℤ, +𝑝⟩𝑖𝑠𝑑𝑒𝑛𝑜𝑡𝑒𝑑𝑏𝑦𝑍𝑝

Binary operation table for

International Standard Book Number: (ISBN) &
Modular arithmetic
• Books are identified by a ten digit number, abbreviated by
ISBN.
• For example, the book Field and Galois Theory, published by
Springer, has the number ISBN 0-387-94753-1.
• The first digit identifies the language in which the book is
written,
• The second block of digits identifies the publisher,
• The third block identifies the book itself, and
• The final digit is the check digit.

• In this scheme, each digit can be a numeral 0,..., 9 or X, which
represents 10. A ten digit number is a valid ISBN
provided that
is divisible by 11
Hence a valid ISBN number
For the validity 10 .0+9.3+8.8+7.7 +6.9+5.7+4.3+3.2+2.9+? = multiple of 11.
That is 265 +? = (11.24 + 1)+? must be divisible by 11
Therefore ? = 10 = X

Mathematical logic
• What is a Logic ?
• What is a Mathematical Logic?
• Logic (from the Greek "logos", has a meanings including word,
thought, idea, argument, account, reason or principle) is
the study of reasoning, or the study of the principles and criteria
of valid inference and demonstration.
• It attempts to distinguish good reasoning from bad reasoning.
• Mathematical logic is often divided into the fields of set
theory, model theory, recursion theory, and proof theory.

• To build any language we need to have the first alphabet
alphabets , words , statements and valid statements.
• For example we have,
Alphabets Words Language Other languages
அ, ஆ, இ,
ஈ,…
அன் பு தமிழ்
A, B , C ,
D,…
Love English French, German
Αγάπη
Greek
0, 1, 2 , 3,…9 12 15 22 5 Decimal Number Ternary
0, 1 (100101000101011111001)2 Binary Number Hexa decimal
In fact Mathematically speaking , every language, built on a declared
alphabet sets w.r.to the binary operation “ concatenation” is a Monoid.

Consider the following phrases in english/தமிழ்
1. Apple is a fruit
2. Bharathiyar is a novalist
3. கதவை மூடவும்
4. Drop it !
5. முதுமவையிை் யாவையய
கிவடயாது
6. Wow! beautiful
7. 17+29 = 0
8. 2+7 > 5
9. நாை
் ச ாை்ைசதை்ைாம்
ச ாய்
1. True
2. False
3. Command
4. Command
5. False
6. Exclamation!
7. Relatively true
8. True
9. Paradox
truth value can be assigned are the needed “ Statement “

• “ Statements “ for which a definite truth value could be
assigned is called a proposition .
• which do not have any connecting words like and , or
, because, since… are called simple or atomic
statement other wise it is called compound statements.
• A statement formula is an expression involving one or
more statements connected by some logical
connectives.

• Depending on the language skills of persons same
statements can be stated with different words. But a
machine cannot understand all the possible equivalent
statements in general.
• Hence there is a need to symbolise the atomic
/simple propositions, using symbols /english alphabets like
p, q, r…. and the connecting words as Connectives so that
the entire language can be programmed easily .
• In view of this the logic we study is also called a symbolic
logic.

(i) 17 is a prime number and it is raining outside therefore
government teachers get the 5 % Dearness allowance.
Symbolise the atomic statements of the following sentences
p: 17 is a prime number
q: It is raining out side
r : Government teachers will get dearness allowance
(ii) 17 is a prime number and it is raining outside therefore
government teachers get the 5 % Dearness allowance.

24
Sl.No Name symbol English name
1
Negation
Not
2 Conjunction And
3 Disjunction Or
4 Conditional If...then...
5 Bi-conditional If and only if
Connectives

Truth table
T F
F T
for negation
for conjunction
T T T
T F F
F T F
F F F
for disjunction
T T T
T F T
F T T
F F F
for conditional
T T T
T F F
F T T
F F T
for bi -conditional
T T T
T F F
F T F
F F T
for Exclusive OR
T T F
T F T
F T T
F F F

Truth table contd…
for Nand
T T F
T F T
F T T
F F T
𝑝 ↑ 𝑞 = ¬(𝑝 ∧ 𝑞)

Truth table contd… for Nor
T T F
T F F
F T F
F F T
𝑝 ↓ 𝑞 = ¬(𝑝 ∨ 𝑞)

Construct the truth table for the statement
formula
(𝑝 ∨ 𝑞) ∧ (𝑝 ∨ ¬𝑞)
What ever be the truth value compositions of
‘p’ and ‘q’ the truth value of the given statement
formula is False.

Construct the truth table for the statement
formula
((𝑝 ⟶ 𝑞) ∧ 𝑝) ⟶ 𝑞
What ever be the truth value compositions of
‘p’ and ‘q’ the truth value of the given statement
formula is True

(i) 19 is not a prime number and all the angles of a triangle are equal.
(ii) 19 is a prime number or all the angles of a triangle are not equal
(iii) 19 is a prime number and all the angles of a triangle are equal
(iv) 19 is not a prime number
Symbolise the following statements
Let
p: 19 is a prime number
q: all the angles of a triangle are equal
(𝑖)¬𝑝 ∧ 𝑞 (𝑖𝑖)𝑝 ∧ ¬𝑞
(𝑖𝑖𝑖)𝑝 ∧ 𝑞 (𝑖𝑣)¬𝑝

Classification of Statements
A statement is said to be a tautology if its
truth value is always T irrespective of the truth
values of its component statements. It is denoted
by 𝕋. It will be of the form , for some statement P
𝑃 ∨ ¬𝑃
A statement is said to be a contradiction if its
truth value is always F irrespective of the truth
values of its component statements. It is denoted
by F . It will be of the form , for some statement P
𝑃 ∧ ¬𝑃
A statement which is neither a contradiction nor a tautology is called a
contingency .
Note: In order to write the truth table for a given statement formulae
the number of rows needed are
where ’n’ is the number of atomic/simple statements.
2𝑛

Equality of statements
Let P(p, q, r,), Q( p ,q, r ,…) be any two statements with same
simple statement variables . We say these two statements are
equivalent if the the truth values of these two statements are the same
irrespective of the truth values their components. In other words the
last column of P(p, q, r,…), Q( p ,q, r ,…) are the same. For example
𝑝 ⟶ 𝑞𝑎𝑛𝑑¬𝑝 ∨ 𝑞𝑎𝑟𝑒𝑒𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡
T T T T
T F F F
F T T T
F F T T

Show the equivalence with and without the use of truth
table
Solution with out truth table

Solution with truth table
p q r q—-> r p—->(q—->r) p^q (p^q) —->r
T T T T T T T
T T F F F T F
T F T T T F T
T F F T T F T
F T T T F
F T F F F
F F T T
F F F T
You may try it as an exercise
for other truth values

Consequences of conditional statement
𝐶𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛𝑎𝑙𝑠𝑡𝑎𝑡𝑒𝑚𝑒𝑛𝑡𝑝 ⟶ 𝑞
𝐶𝑜𝑛𝑣𝑒𝑟𝑠𝑒𝑠𝑡𝑎𝑡𝑒𝑚𝑒𝑛𝑡𝑞 ⟶ 𝑝
𝐼𝑛𝑣𝑒𝑟𝑠𝑒𝑠𝑡𝑎𝑡𝑒𝑚𝑒𝑛𝑡¬𝑝 ⟶ ¬𝑞
𝐶𝑜𝑛𝑡𝑟𝑎𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒𝑠𝑡𝑎𝑡𝑒𝑚𝑒𝑛𝑡¬𝑞 ⟶ ¬𝑝
𝑝 ⟶ 𝑞𝑎𝑛𝑑¬𝑞 ⟶ ¬𝑝𝑎𝑟𝑒𝑒𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡
𝑝 ⟶ 𝑞 = ¬𝑞 ∨ 𝑝 = 𝑝 ∨ ¬𝑞
¬𝑞 ⟶ ¬𝑝 = ¬¬𝑝 ∨ ¬𝑞 = 𝑝 ∨ ¬𝑞

Duality statements

Dual of a statement
Let Q( p ,q, r ,…) be any statement or statement formula with
statement variables p, q, r ,… . The dual of Q is a statement say,
𝑄′
(𝑝, 𝑞, 𝑟) is statement obtained from Q by replacing
∨ 𝑏𝑦 ∧,∧ 𝑏𝑦 ∨, 𝑇𝑏𝑦𝐹, 𝐹𝑏𝑦𝑇
Duality law
𝑇ℎ𝑒𝑑𝑢𝑎𝑙𝑄′
(𝑝, 𝑞, 𝑟)𝑜𝑓𝑄(𝑝, 𝑞, 𝑟)𝑖𝑠
¬𝑄(¬𝑝, ¬𝑞, ¬𝑟)

Considering the negation , conjunction and disjunction as
operators on the set of statement formulae , say S , we have
⟨𝑆,∧⟩𝑎𝑛𝑑⟨𝑆,∨⟩ are algebraic systems.
also these operators satisfies associative , commutative, distributive laws.
(𝑝 ∨ 𝑞) ∨ 𝑟 = 𝑝 ∨ (𝑞 ∨ 𝑟); 𝑝 ∧ (𝑞 ∧ 𝑟) = (𝑝 ∧ 𝑞) ∧ 𝑟
𝑝 ∨ 𝑞 = 𝑞 ∨ 𝑝; 𝑝 ∧ 𝑞 = 𝑞 ∧ 𝑝
𝑝 ∧ (𝑞 ∨ 𝑟) = (𝑝 ∧ 𝑞) ∨ (𝑝 ∧ 𝑟); 𝑝 ∨ (𝑞 ∧ 𝑟) = (𝑝 ∧ 𝑞) ∨ (𝑝 ∧ 𝑟

S.No Web Link For what
1
http://www-history.mcs.st-
and.ac.uk/Day_files/Year.html
To Know the mathematicians
of the day
2 http://www.wolframalpha.com/
To compute every thing in
mathematics
3
https://www.geeksforgeeks.org/p
roposition-logic/
for Mathematics logic
4 https://ocw.mit.edu/index.htm MIT Open course ware

e-mail: drtns2008@gmail.com
If you wish to contact me
you may do so through
Face book : Tirunelveli Nellaiappan Shanmugam
FB Group : Higher Secondary Mathematics Tamizh
Nadu

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பிரி
யைாம்
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