DISMATH_Part2

DISMATH
Discrete Mathematics
Methods of Proof, Algorithms,
and Number Theory
De La Salle University
September 2014

Overview
- Methods of Proof
- Algorithms

Overview
- Methods of Proof
- Algorithms
- The Growth of Functions

Overview
- Methods of Proof
- Algorithms
- Complexity of Algorithms

Overview
- Methods of Proof
- Algorithms
- Integers and Division

Overview
- Methods of Proof
- Algorithms
- Integers and Division
- Number Theory and Applications

Why Study Proofs?
Computing systems are doing so much:
How can we guarantee they work?

Why Study Proofs?
Why not just testing?
- Integrates well with programming
- No new languages, tools required
- Conclusive evidence for bugs

Why Study Proofs?
Why not just testing?
- Integrates well with programming
- No new languages, tools required
- Conclusive evidence for bugs
Because. . .
- Diﬃcult to assess coverage
- Cannot demonstrate absence of bugs
- No guarantees for safety-critical systems

Why Study Proofs?
Formal Veriﬁcation
1. SOFTWARE:
If you want to debug a program beyond a doubt,
prove that it’s bug-free! Deduction and proof
provide universal guarantees.
2. HARDWARE:
Proof-theory has recently also been shown to be
useful in discovering bugs in pre-production
hardware.

Methods of Proof
- Direct Proof

Methods of Proof
- Direct Proof
- Proof by Contraposition
(Indirect)

Methods of Proof
- Direct Proof
(Indirect)
- Vacuous and Trivial Proof

Methods of Proof
- Direct Proof
(Indirect)
- Proof by Contradiction (Indirect)

Methods of Proof
- Direct Proof
(Indirect)
- Proof by Contradiction (Indirect)
- Proof by Equivalence

Direct Proof
- In a conditional statement p → q, assume that
p is true and use deﬁnitions and previously
proven theorems, to show that q must also be
true.

Direct Proof
- In a conditional statement p → q, assume that
p is true and use deﬁnitions and previously
proven theorems, to show that q must also be
true.
- Ex. Give a direct proof of the theorem:
"If n is an odd integer, then n2
is odd."

Proof by Contraposition
- We take ¬q as a hypothesis, and using
deﬁnitions, and previously proven theorems, we
show that ¬p must follow.

Proof by Contraposition
- We take ¬q as a hypothesis, and using
deﬁnitions, and previously proven theorems, we
show that ¬p must follow.
- Ex. Prove that if n is an integer and 3n + 2 is
odd, then n is odd.

Exercise
- Prove that if n = ab, where a and b are positive
integers, then a ≤
√
n or b ≤
√
n.

Vacuous and Trivial Proofs
- Vacuous proof
Show that p is false, because p → q must be
true when p is false.
- ¬p → (p → q)

Vacuous and Trivial Proofs
- Vacuous proof
Show that p is false, because p → q must be
true when p is false.
- ¬p → (p → q)
- Trivial proof
Show that q is true, it follows that p → q must
also be true.
- q → (p → q)

Exercise
- Prove the statement: If there are 30 students
enrolled in this course this semester, then
62
= 36.

Exercise
- Prove the statement: If there are 30 students
enrolled in this course this semester, then
62
= 36.
- Prove the statement. If 6 is a prime number,
then 62
= 30.

Exercise
- Prove that if n is an integer and n2
is odd, then
n is odd.

Proof by Contradiction
- Show that assuming ¬p is true leads to a
contradiction.

Proof by Contradiction
- Show that assuming ¬p is true leads to a
contradiction.
- Ex. Prove that
√
2 is irrational by giving a
proof by contradiction.

Exercise
- Give a proof by contradiction of the theorem
If 3n + 2 is odd, then n is odd.

Proof by Equivalence
- To prove a theorem that is a biconditional
statement, p ↔ q, we show that p → q and
q → p are both true.

Proof by Equivalence
- To prove a theorem that is a biconditional
statement, p ↔ q, we show that p → q and
q → p are both true.
- (p ↔ q) ↔ [(p → q) ∧ (q → p)]

Exercise
- Prove the theorem
If n is a positive integer, then n is odd if and
only if n2
is odd.

Exercise
- Show that these statements about the integer n
are equivalent:
P1 : n is even.
P2 : n – 1 is odd.
P3 : n2
is even.

Exercise
- T/F: Every positive integer is the sum of the
squares of two integers.

Algorithm
- Algorithm
a ﬁnite set of precise instructions for
performing a computation or for solving a
problem.

Algorithm
- Algorithm
a finite set of precise instructions for
performing a computation or for solving a
problem.
- Ex. Describe an algorithm for finding the
maximum (largest) value in a finite sequence of
integers.

Finding the Maximum Element
Algorithm
1. Set the temporary maximum equal to the
ﬁrst integer in the sequence.

Algorithm
2. Compare the next integer in the sequence to
the temporary maximum, if larger, set the
temporary maximum equal to this integer.

Algorithm
3. Repeat the previous step if there are more
integers in the sequence.

Algorithm
3. Repeat the previous step if there are more
integers in the sequence.
4. Stop when there are no integers
left in the sequence.

Pseudocode
- high-level description of an algorithm that uses
the structural conventions of a programming
language, but is intended for human reading.

Pseudocode
- high-level description of an algorithm that uses
the structural conventions of a programming
language, but is intended for human reading.
- computer programs can be produced in any
computer language using the pseudocode
description as a starting point.

Pseudocode
PROCEDURE max(a1, a2, . . ., an : integers)
max = a1
for i = 2 to n
if max aj then max = aj
{Output: max is the largest element}

Preconditions and Postconditions
- Preconditions
statements that describe valid input

- Preconditions
- Ex. (a1, a2, . . ., an) ∈ Z

- Preconditions
- Ex. (a1, a2, . . ., an) ∈ Z
- Postconditions
conditions that the ouput should satisfy when
the program has run

- Preconditions
- Ex. (a1, a2, . . ., an) ∈ Z
- Postconditions
conditions that the ouput should satisfy when
the program has run
- Ex. Output: max is the largest element

Properties of Algorithm
- Input
An algorithm has input values from a speciﬁed
set.

- Input
set.
- Output
From each set of input values an algorithm
produces output values from a speciﬁed set.

- Input
set.
- Output
From each set of input values an algorithm
produces output values from a specified set.
- Definiteness
The steps of an algorithm must be defined
precisely.

- Correctness
An algorithm should produce the correct
output values for each set of input values.

- Correctness
- Finiteness
An algorithm should produce the desired
output after a ﬁnite number of steps.

- Correctness
- Finiteness
An algorithm should produce the desired
output after a ﬁnite number of steps.
- Generality
The procedure should be applicable for all
problems of the desired form, not just for a
particular set of input.

Algorithm Sample Program

Searching Algorithms
- The problem of locating an element in an
ordered list.

ordered list.
- Locate an element x in a list of distinct
elements a1, a2, . . ., an, or determine that it is
not in the list.

ordered list.
- Locate an element x in a list of distinct
elements a1, a2, . . ., an, or determine that it is
not in the list.
- Solution: the location of the term in the list
that equals x (that is, i is the solution if x = ai
) and is 0 if x is not in the list.

Linear Search Algorithm
PRECONDITION: Linear search (x : integer, a0, a1,. . . , an :
distinct integers)
- 1. Compare x and a0. If x = a0, location = 0,
else proceed to next element.
POSTCONDITION: Output the location.

distinct integers)
- 2. Repeat step 1 while a match has not been
found and there are still elements.

distinct integers)
- 3. Output the location if a match is found, else
location = -1 signifying not found.

Linear Search Pseudocode
PROCEDURE linear search
(PRECONDITION: x : integer, a0, a1, . . ., an–1 :
distinct integers)
i = 0
while (i n and x = ai )
i = i + 1
if i n then location = i
else location = –1
{POSTCONDITION: location is the subscript of
the term that equals x , or is –1 if x is not found}

Binary Search Algorithm
PRECONDITION: Binary search (x : integer, a0, a1,. . . , an–1 :
sorted integers)
- 1. Compare x to the middle term of the list. If
x is larger, choose the upper half, else choose
the lower half.

sorted integers)
the lower half.

sorted integers)
the lower half.
- 3. Output the location if a match is found, else
location = -1 signifying not found.

Example
- Search for 19 in the list
1 2 3 5 6 7 8 10 12 13 15 16 18
19 20 22

Binary Search Pseudocode
sorted integers)
i = 0 {i is left endpoint of search interval}
j = n – 1 {j is right endpoint of search interval} while i j
{
m = (i + j)/2
if x am then i = m + 1
else j = m
}
if x = ai then location = i
else location := –1

Sorting Algorithms
- The problem of putting elements in increasing
order.

Sorting Algorithms
order.
- Given a list of elements of a set, sort in
increasing order.

Sorting Algorithms
order.
- Given a list of elements of a set, sort in
increasing order.
- Solution: bubble sort, insertion sort, etc.

Bubble Sort Algorithm
PRECONDITION: Bubble Sort (a1, a2,. . . , an : real numbers
with n ≥ 2)
- 1. Successively compare adjacent elements.
POSTCONDITION: Output the elements a1, a2,. . . , an
in increasing order.

with n ≥ 2)
- 2. Interchange elements if they are in the
wrong order.

with n ≥ 2)
wrong order.
- 3. Repeat until there are elements.

with n ≥ 2)
wrong order.
- 4. Output the list of elements in increasing
order.

Bubble Sort Pseudocode
PROCEDURE bubble Sort
(PRECONDITION: a1, a2,. . . , an : real numbers
with n ≥ 2)
for i = 1 to n – 1
for j = 1 to n – i
if aj aj+1
then interchange aj and aj+1
{POSTCONDITION: Output the elements a1,
a2,. . . , an in increasing order.}

Insertion Sort Algorithm
PRECONDITION: Insertion Sort (a1, a2,. . . , an : real numbers
with n ≥ 2)
- 1. Compare second element a2 with the ﬁrst
element a1.

with n ≥ 2)
element a1.
- 2. If a2 is smaller, place it before a1 else place
it after a1.

with n ≥ 2)
element a1.
it after a1.

with n ≥ 2)
element a1.
it after a1.
- 4. Output the list of elements in increasing
order.

Insertion Sort Pseudocode
PROCEDURE Insertion Sort
PRECONDITION: a1, a2,. . . , an : real numbers with n ≥ 2
for j = 2 to n
{
i = 1
while aj ai
i = i + 1
m = aj
for k = 0 to j – i – 1
aj–k = aj–k–1
ai = m
}
{POSTCONDITION: Output the elements a1, a2,. . . , an
in increasing order.}

Greedy Algorithm
- selects the best choice at each step, instead of
considering all sequences of steps that may lead
to an optimal. solution.

Greedy Algorithm
- selects the best choice at each step, instead of
considering all sequences of steps that may lead
to an optimal. solution.
- applied in optimization problems where a
solution to the given problem either minimizes
or maximizes the value of some parameter.

Exercise
- Consider the problem of making n cents change
with quarters, dimes, nickels, and pennies, and
using the least total number of coins.

Greedy Change-Making Algorithm
Pseudocode
PROCEDURE change
PRECONDITION: c1, c2,. . . , cn values of denominations of
coins, where c1 c2 . . . cn; n ∈ Z+
for i = 1 to r
while n ≥ ci
add a coin with value ci to the change
n = n – ci
endwhile
endfor
{POSTCONDITION: Output the minimum number of coins.}

Growth of Functions
- The growth of functions is often described
using Big-O Notation.
The constants C and k in the deﬁnition of big-O notation are
called witnesses.

Growth of Functions
- The growth of functions is often described
using Big-O Notation.
- Deﬁnition: Let f and g be functions from
R → R; f (x) is O(g(x)) if there are constants
C and k such that
|f (x)| C |g (x)|
whenever x k.
The constants C and k in the deﬁnition of big-O notation are
called witnesses.

Example
- Show that f (x) = x2
+ 2x + 1 is O(x2
).
◦ A useful approach for ﬁnding a pair of witnesses
is to ﬁrst select a value of k for which the size of
|f (x)| can be readily estimated when x k.

Drill
- Show that 7x2
is O(x3
).

Drill
- Show that 7x2
is O(x3
).
- Show that n2
is not O(n).

Drill
- How can big- O notation be used to estimate
the sum of the ﬁrst n positive integers?

Drill
- How can big- O notation be used to estimate
the sum of the ﬁrst n positive integers?
- Give big- O estimates for the factorial function
and the logarithm of the factorial function.

Big-Omega and Big-Theta Notation
- Big-O notation does not provide a lower bound
for the size of f (x).

- For the lower bound, we use big-Omega
(big-Ω) notation.

- For the lower bound, we use big-Omega
(big-Ω) notation.
- For the both lower and upper bound, we use
big-Theta (big-Θ) notation.

Algorithm Time Complexity
- can be expressed in terms of the number of
operations used by the algorithm when the
input has a particular size.

Algorithm Time Complexity
- can be expressed in terms of the number of
operations used by the algorithm when the
input has a particular size.
- the number of comparisons will be used as the
measure of the time complexity of the
algorithm, because comparisons are the basic
operations used.

Example
- Describe the time complexity of algorithm for
ﬁnding the maximum element in a set.
PROCEDURE max(a1, a2, . . ., an : integers)
max = a1
for i = 2 to n
if max aj then max = aj
{Output: max is the largest element}

Drill
- What is the worst-case complexity of the
bubble sort in terms of the number of
comparisons made?
PROCEDURE bubble Sort
(PRECONDITION: a1, a2,. . . , an : real numbers
with n ≥ 2)
for i = 1 to n – 1
for j = 1 to n – i
if aj aj+1
then interchange aj and aj+1
{POSTCONDITION: Output the elements a1,
a2,. . . , an in increasing order.}

Division and Modulo Operator
- Let a be an integer and d a positive integer.
Then there are unique integers q and r, with
0 ≤ r d, such that a = dq + r.

q = a div d r = a mod d

q = a div d r = a mod d
- In the equality given, d is called the divisor, a
is called the dividend, q is called the quotient,
and r is called the remainder.

Example
- What are the quotient and remainder when 101
is divided by 11 ?

Example
- What are the quotient and remainder when 101
is divided by 11 ?
- What are the quotient and remainder when –11
is divided by 3?

Modulo Equivalence
- If a and b are integers and m is a positive
integer, then a is congruent to b modulo m if m
divides a – b.

Modulo Equivalence
- If a and b are integers and m is a positive
integer, then a is congruent to b modulo m if m
divides a – b.
- We use the notation a ≡ b( mod m) to indicate
that a is congruent to b modulo m.
a ≡ b mod m iﬀ. m|(a – b)

Example
- Determine whether 17 is congruent to 5
modulo 6 and whether 24 and 14 are congruent
modulo 6.

Applications
- Cryptology: the study of secret messages.
- Ex. Caesar Cipher: Messages are made secret
by shifting each letter three letters forward in
the alphabet.

Applications
- Cryptology: the study of secret messages.
- Ex. Caesar Cipher: Messages are made secret
by shifting each letter three letters forward in
the alphabet.
- Caesar’s encryption method can be represented
by the function f that assigns to the
nonnegative integer p, p ≤ 25, the integer f(p)
in the set {0, 1, 2, ..., 25} with
f (p) = (p + 3) mod 26

Example
- What is the secret message produced from the
message MEET YOU IN DLSU using the
Caesar cipher?

Example
Caesar cipher?
- PHHW BRX LQ GOVY

Example
Caesar cipher?
- PHHW BRX LQ GOVY
f –1
(p) = (p – 3) mod 26

Representation of Integers
- Let b be a positive integer greater than 1.
Then if n is a positive integer, it can be
expressed uniquely in the form
n = akbk
+ ak–1bk–1
+ ... + a1b + a0

Representation of Integers
- Let b be a positive integer greater than 1.
Then if n is a positive integer, it can be
expressed uniquely in the form
n = akbk
+ ak–1bk–1
+ ... + a1b + a0
- where k is a nonnegative integer, a0, a1, . . . , ak
are nonnegative integers less than b, and
ak = 0.

Example
- What is the decimal expansion of the integer
that has (101011111)2 as its binary expansion?

Example
- What is the decimal expansion of the integer
that has (101011111)2 as its binary expansion?
- What is the decimal expansion of the
hexadecimal expansion of (2AEOB)16?

Base Conversion Algorithm
1. Divide n by b to obtain a quotient and
remainder, n = bq0 + a0, 0 ≤ a0 b

2. The remainder, a0, is the rightmost digit in
the base b expansion of n.

3. Divide q0 by b to obtain q0 = bq1 + a1,
0 ≤ a1 b

0 ≤ a1 b
4. The remainder, a1, is the second digit from
the right in the base b expansion of n.

0 ≤ a1 b
4. The remainder, a1, is the second digit from
the right in the base b expansion of n.
5. Continue process until we obtain a quotient
equal to zero.

Example
- Find the base 8, or octal, expansion of
(12345)10.

Example
(12345)10.
- Find the hexadecimal expansion of (177130)10.

Example
(12345)10.
- Find the hexadecimal expansion of (177130)10.
- Find the binary expansion of (241)10.

Base b Expansions Algorithm
Pseudocode
PROCEDURE base b expansion
PRECONDITION: (n: positive integer);
q = n
k = 0
while q = 0
{
ak = qmodb
q = q/b
k = k + 1
}
{POSTCONDITION: the base b expansion of n is
(ak–1, . . . , a1, a0)b}

Euclidean Algorithm
A method for computing the greatest common
divisor of two integers.
- Let a = bq + r, where a, b, q, and r are
integers. Then gcd(a, b) = gcd(b, r).

Euclidean Algorithm
gcd(a, b) = gcd(ro, r1) = gcd(r1, r2) = . . .
= gcd(rn–2, rn–d = gcd(rn1
, rn) = gcd(rn, 0) = rn.

Euclidean Algorithm
gcd(a, b) = gcd(ro, r1) = gcd(r1, r2) = . . .
= gcd(rn–2, rn–d = gcd(rn1
, rn) = gcd(rn, 0) = rn.
- The greatest common divisor is the last
nonzero remainder in the sequence of divisions.

Example
- Find the greatest common divisor of 414 and
662 using the Euclidean algorithm.

Euclidean Algorithm Pseudocode
PROCEDURE GCD
PRECONDITION: (a, b: positive integers; ab);
x = a
y = b
while y = 0
{
r = x mod y
x = y
y = r
}
{POSTCONDITION: GCD(a, b) is x }

Reference:
Rosen, K.H. Discrete Mathematics and Its
Applications (7 ed.), New York, McGraw-Hill

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