ece math 311_topic 5 to 9

8/13/2019 Ece Math 311_topic 5 to 9

1/75

DISCRETE MATHEMATICS

ECE MATH 311

TOPIC 5:

PROOFS APPLICATIONS ONSETS & SUMMATION

By: Edison A. Roxas, MSECE


2/75

Proofs Application on Sets & Summation 2

OBJECTIVES

At the end of the topic, the students should be able to:

1. Understand the ideas of sets and summation;

2. Define the characteristics of a set;

3. Analyze set representation, laws and logic;

4. Solve problems using set operations and Venn diagram;

and

5. Analyze problems involving summation and products .

earoxas @ UST 2013


3/75

SETS

It is a well defined collection of distinct objects.

Well defined set means that it is possible to

determine whether an object belongs to a given

set.

The objects are called members and elements.

denotes element of a set.

aS is read as a is not an element of set S.

earoxas @ UST 2013 3Proofs Application on Sets & Summation


4/75

EXAMPLES

1. Collection of vowels in the English Alphabet:

a, e, i, o, u

2. Collection of odd numbers: 1, 3, 5, 7

3. Collection of Laptop brands: Asus, Acer,

Toshiba, Sony

4. Collection of favorite colors.5. Collection of good movies.



5/75

SET REPRESENTATION

Two Ways of Describing a Set:

1. Tabular / Roster Form = a method ofdescribing a set where elements are

separated by commas and enclosed bybraces.

2. Rule Form = is a method of describing a set

which makes use of the description {x|} andis read as the set of all elements x such thatx.



6/75

EXAMPLE 5.1:

Write the corresponding Rule or Roster form

given the following:

1. {x| x is an odd number between 0 and 9}

2. {x| x2 1| 0 x 5, x Z}

3. {2, 3, 5, 7, 11, 13, 17}



7/75

KINDS OF SETS

1. Null or Empty Set { }, contains no

element.

2. Equal Sets Sets A and B are equal,

denoted by A = B. A = {1,2,3} and B = {2,1,3}

are equal sets.

3. Equivalent Sets denoted by A~B, is they

have the same number of elements.

example: C = {a,b,c} and D = {4,5,6}



8/75

KINDS OF SETS

4. Finite Sets contains countable number of

elements.

5. Infinite Sets if the counting elements has

no end. The sets of integer Z, positive integers

N (or natural numbers), negative integers Z- ,

and non negative integers (or whole numbers)

are infinite sets.



9/75

KINDS OF SETS

6. Universal Sets is the totality of elements

under consideration.

7. Joint Sets are sets that have common

elements.

8. Disjoint Sets are sets that have no

common element. The set {0} are also disjoint

sets.



10/75

FRACTIONS & DECIMALS

Rational Number denoted by Q may

either be a fraction or integer. It is a number

in the form of the ratio of two integers a and

b denoted by a/b, where b 0.

Fractions can be expressed as a form of

terminating decimal or repeating non-

terminating decimals.



11/75

FRACTIONS & DECIMALS

Terminating Decimals:

a. Terminating Decimals: = 0.5; 1/5 = 0.2

b. Repeating nonterminating decimal:

1/3 = 0.333 ; 2/3 = 0.6666c. Nonrepeating nonterminating decimal:

pi = 3.14159265

e = 2.718281828

The third form of decimals is called irrationalnumbers.



12/75

EXAMPLE 5.2: SET OPERATIONS

Given: u = {2, 4, 6, 8, 10, 12}

A = {2, 4, 10} B = {6, 10, 12}

1. Union of Sets A and B2. Intersection of Sets A and B

3. Complement of A

4. Difference of Sets A and B



13/75

LAW OF SETS

1. Commutative Laws

2. Associative Laws

3. Identity Laws

A U 0 = A A u = A

4. Inverse / Complement Laws

A U A = u A A = 0

5. Distributive Laws

6. De Morgans Laws



14/75

VENN DIAGRAM

The Venn Diagram of sets makes use of a

rectangle representing the universal set and

circles are subset which may be shaded under

consideration.



15/75

EXAMPLE 5.3: VENN DIAGRAM

In a marketing survey conducted involving 150 companies, it wasfound out that

70 uses Brand A

75 uses Brand B

95 uses Brand C

30 uses Brands A and B

45 uses Brands A and C

40 uses Brands B and C

10 uses Brands A, B and C

What is the number of companies that did not purchased anybrand?



16/75

INTERVALS ON NUMBER LINE & SUBSETS

The geometric interpretation to the set if real

numbers is done by associating them with

points on the horizontal line (or real line or x

axis), called the number line.

The real number line will be used in forming

intervals. These intervals may be open (a,b),

closed [a,b], or half open, or half closed[a,b) or (a,b].

earoxas @ UST 2013 Proofs Application on Sets & Summation 16


17/75

UNBOUNDED OR INFINITE INTERVALS

Intervals on a number line of the form:

a. [a, + inf)

b. (a, + inf)c. (-inf, a]

d. (-inf, a)

e. (-inf, +inf)



18/75

SUMMATION & MULTIPLICATION SYMBOLS

Subscript notations are used when we aredealing with a large collection of objects.

Summation can be used to represent a

polynomial. The Greek sigma is used tostands for summation, .

In analogy with the symbol , the Greek pi

is generally used for the product sign. Thisproduct sign may also be used in compactform using the factorial notation n!.



19/75

UNIVERSAL GATES

- Implementation of the Logic Gates using NAND

and NOR gates.



20/75

EXAMPLES

1. Given: A = {1, 7, 9, 10} & B = {4, 7, 9, 11}

Find : a. A B b. B A

2. Find the actual expanded expression:a. F(A, B, C) = BC + A(C+B)

b. F(X, Y, Z) = (X + Y) (Y + Z)

3. Prove that (2, 3, 4, 5) = (0, 1, 6, 7) of the F(x, y, z).



21/75

EXAMPLES

4. Prove:

The Uniqueness of Complement Theorem

Given A and B as subsets of Universe, then B = A; if

and only if A U B = U and A B =0.

Use:

Definition: U = A + ATheorem: A U A = U

A B = 0



22/75


ECE MATH 311

TOPIC 6:

PROOFS AND RULES OF

INFERENCE

By: Edison A. Roxas


23/75

MATHEMATICAL SYSTEM

A Mathematical System consists of

Axioms, Definitions, and Undefined Terms.

Definitions = used to create new

concepts in terms of existing ones.

Axioms =Also called Postulates, are

statements that we assume to be TRUE.

* Some terms are not explicitly defined but

implicitly defined in an axiom.

Proofs and Rules of Inferenceearoxas @ UST 2013


24/75

MATHEMATICAL SYSTEM

Theorem = a proposition that have beenproven TRUE.

Lemma A theorem that is not too

interesting on its own but is useful inproving another theorem.

Corollary A theorem that follows

from another theorem.Proof =An argument that establishes thetruth of a theorem.



25/75

MATHEMATICAL SYSTEM

EXAMPLES: Real Numbers

Axioms:

For all Real Numbers x AND y, xy = yx.

There is a subset P of Real Numbers

Satisfying:- If x and y are in P, then x + y andxy are in P.

- If x is a real number, then

exactly one of the statements isTRUE.

x is in P, x = 0 , -x is in P.



26/75

MATHEMATICAL SYSTEM


Definitions:

The element in P are called positiveReal Numbers.

The absolute value /x/ of a real number

x is defined to be x if x is positive or 0

and x otherwise.



27/75

MATHEMATICAL SYSTEM


Theorems:

x . 0 = 0; for every real numberFor all real numbers, x, y and z, if x y

and y z, then x z.



28/75

Lemma:

If n is a Positive Integer, then either n 1is a positive integer or n 1 = 0.

MATHEMATICAL SYSTEM




29/75

DIRECT PROOF

Assumes that P(x) is TRUE then using P(x)

as well as other axioms, definitions and

previously derived theorems, shows

directly that Q(x) is TRUE.



30/75

Example:

Definition:

An integer is even if there is an integer k

such that n = 2k.

An integer is odd if there is an integer k such

that n = 2k + 1.

Theorem:

For all integers m and n, if m is ODD and n

is EVEN, then m + n is ODD.



31/75

VACUOUS PROOF AND

TRIVIAL PROOF In conditional statements we know that pq is

TRUE when p is FALSE and call it TRUE by default

of vacauously true.

Consequently, if we can show that p is FALSE, thenwe have vacuous proof of the conditional statement

pq.

We can also prove a conditional statement if we

know that q is TRUE, by showing that p is TRUE, itfollows pq must also be TRUE.

A proof that uses the fact that q is TRUE to prove

pq to be TRUE is called a TRIVIAL PROOF.



32/75

MODUS PONENS

The First Rule of Inference is calledModus Ponens or the Rule of Detachment.

It comes from Latin translated as the

Method of Affirming. The symbolic form is given as:

[p^(pq)] q ; it is written in the tabular

form ppq

therefore q



33/75

MODUS PONENS

Example:

If it is snowing today, then we will go

skiing.It is snowing today.

Therefore,

We will go skiing.



34/75

LAW OF SYLLOGISM

The second rule of inference is given by

the logical implication:

[(p q)^(q r)] (p r)

where p, q, and r are any statements. In

tabular form it is written

pq

qr

therefore pr



35/75

Example:

If it is sunny, then I will not bring an

umbrella. If I have no umbrella, then I will

visit a friend. If it is sunny then I will visit a

friend.

LAW OF SYLLOGISM



36/75

The Third Rule of Inference is called

Modus Tollens.

Modus Tollens comes from Latin and can

be translated as method of denying.

[(p q)^ ~q] ~p

pq

~q

therefore ~p

MODUS TOLLENS



37/75

MODUS TOLLENS

Example:

If Claire is elected president of Math Club,

then Jacob will be a member of the club.

Jacob did not wish to be a member of the

club.

Therefore Claire was not elected as

president of the Math Club.



38/75


ECE MATH 311

TOPIC 7:

RELATIONS AND FUNCTIONS

By: Edison A. Roxas


39/75

OBJECTIVES

At the end of the topic, the students should be ableto:

1. Distinguish a function form from a mere

relation;

2. Find the domain and range of functions or

relations; and

3. Solve problems involving the different

operations on Functions.4. Differentiate the types of functions; and

5. Determine the inverse of a function.

earoxas @ UST 2013 Relations and Functions


40/75

INTRODUCTION TO

RELATION & FUNCTION

The Coolness of Technology

In 80s the concept of a tiny cassette player was

unthinkable. Owning a portable music player with

headphones was the epitome of coolness. This

music player uses tape recording of different songs

in different time duration.

Around two decades later, a compact disk (CD)

replaced the tape player. It contains different songs

which corresponds to the different tracks in the CDwas created. It uses optical light and sensors in

reading this new type of audio recording.



41/75

DEFINITION OF RELATIONS

A relation is the association of different things, objectsor numbers.

A relation may be associated to different objects by one to one, one to many, many to one, ormany to many.

A relation can be described using five different methods: By using arrow diagrams

By using tables

By using ordered pairs

By using graphs By using mathematical sentences and formula.



42/75

DEFINITION FUNCTIONS

A function is a rule or correspondence betweentwo sets; such that each element x in a set 1corresponds to exactly one element in a set 2,called f(x).

Set 1 Domain of the FunctionsDomain is the set of all independentinputs for the functions.

Set 2 Range of the Functions or all possible

values of f(x). Range is the set ofcorresponding values or dependentvalues.



43/75

DEFINITION OF FUNCTIONS

F(x) can be read as f of x; the value of f at

x; or the values of x under f.

A technique that may be used to determine a

function graphically is the presence of asingle arrow that leaves each member of a

domain.

Vertical Line Test = it is used to intersects agraph. If the line intersects a graph more than

once; then the graph is not the graph of a

function.earoxas @ UST 2013 Relations and Functions


44/75

Examples

1. Find the domain and range of the

following functions:

a. y = x1/2

b. y = x2

c. F(X) = X3

d. f(x) = |x|e. y = (x 2) / (x + 1)



45/75

Examples

2. Study and plot the graph given the pointsbelow and answer the questions thatfollows: E(1,7), C(5, - 6), R(0,5), J(-4,9)

a. What is the abscissa of point, threeunits to the left of point J?

b. Determine the coordinates of the point

located two units below and four units tothe right of point E.

c. In what quadrant is point C located?



46/75


47/75

FUNCTIONS

Functions are sometimes called mappings

or transformations.

If f is a function from A to B; A is the domain

of f and B is the codomain of f.

If f(a) = b; b is the image and a is the

preimage of b.

This may also be read as f maps A to B.



48/75

OPERATION ON FUNCTIONS

1. f(x) + g(x) = (f+g)(x)

2. f(x) g(x) = (f g)(x)

3. f(x) . g(x) = (f.g)(x)4. f(x) / g(x) = (f/g)(x)

5. f(x) g(x) = f(g(x))

6. g(x) f(x) = g(f(x))



49/75

Examples on Functions

1. Given: f:x

2x + 3.Find:

a. f(0) c. f(x -2)

b. f(a+h) d. f(x+h) f(x)

2. If f(x) = 5x 4 and g(x) = 2x x2

Find:

a. f(x) g(x) b. (gf)(-1)

3. Given f(x) = 2x2 1 and g(x) = 2/x, find:

a. (fg)(2) b. (ff)(1)



50/75

Types of Functions

1. Even and Odd Functions:

For every x in the domain of the function f:

a. Even Function: f(-x) = f(x)

b. Odd Function: f(-x) = -f(x)c. Neither if it fails both (a) and (b).

2. Continuous and Discontinuous Functions:

- Functions which are continuous are

represented by graphs which can be traced.



51/75

Types of Functions

3. Increasing and Decreasing Functions:

a. A function f is an increasing function whenfor all a and b in the domain of f, if a


52/75

Types of Functions

4. Piecewise Defined Function:These functions have different output formulas for different

parts of the domain.

5. Equal Functions:

Two functions are said to be equal if and only if:a. f and g have the same domain.

b. f(x) = g(x) for all x in the domain.

6. Periodic Functions:

Some functions have graphs that show a repeating pattern.These are called periodic functions.

examples:

f(x) = f(x + 2)

g(x) = g(x+2)



53/75

Types of Functions

7. One to One Functions:

A function is said to be a one to one function ifdifferent values of x always give different values of f(x).

One way of determining if the function is a one to

one function is by applying the horizontal line test. Thistest states that if a horizontal line cuts the graph in atmost one point, then the function is one to onefunction.

8. Onto Functions:

The function f is onto if the range f is equal to Y, thatis all elements of set Y are used as images.



54/75


55/75

EXAMPLES

3. Determine the plot of the unit step

signal, u(n) given the piecewise linear

function as

u(n) = 1, for n0

0, for n


56/75

LOGIC SIMPLIFICATION,

APPLICATIONS AND DESIGN

BY:

Edison A. Roxas


57/75

PROBLEM 1:

A vast container of chemical fluid is to be mixedusing an electronically controlled stirrer.Design the circuit needed to control the stirrer

based on the following the followingcondition:

It will be on if either valve A or valve B isopen;

It will be open if valve C and either valve Aor valve B is open.


58/75

PROBLEM 2:

Design a car alarm circuit that will be ONfollowing the conditions set by the manufactureras:

the engine is ON but the driver Is not wearinghis seatbelt;

the engine is ON but one of the doors of the

car is open; the allowed vehicle payload exceeded its limit.


59/75

PROBLEM 3:

A series of bulb is arranged as A3, A2, A1, and A0;with A3 as the Most Significant Bit (MSB);

design a circuit that will control the output of

the bulb within the values of 0010 and 1000.


60/75

PROBLEM 4:

Simplify the value of the following expression onsets by proving Logically and Mathematically:

Given:

Z (universe) = {1, 2, 3, , 10}

X = {2, 5, 7}

Y = {3, 5, 8, 9}

If:

F = (X Z) U (X Y) U (X Z)


61/75


62/75



63/75


ECE MATH 311

TOPIC 9:

MATRICES

By: Edison A. Roxas, ECE


64/75

MATRIX

A rectangular array of numbers.

The elements are enclosed in square brackets.

Matrices are defined by their dimensions. A

matrix with m ROWS and n COLUMNS is called anm x n matrix.

Two matrices are equal if they have the same

dimensions and every element is the same in

every position.

Matrices are denoted by a boldface capital letter.

earoxas @ UST 2012 Matrices 2


65/75

ADDITION OF MATRICES

Addition or subtraction of matrices

are done on an element to

element basis.Therefore, only matrices of the same

size can be added or subtracted.


EXAMPLE 8 1 Fi d th l f


66/75

EXAMPLE 8.1: Find the value of

z = A + B 2C; given the matrix below:

Say:

1 0 -1A = 2 2 -3

3 4 0

3 4 -1

B = 1 -3 0

-1 1 2

2 3 -1

C = 0 -3 0

-1 2 -2



67/75

MULTIPLICATION OF MATRICES

Let A be an m x n matrix and B an k x l matrix,the product of A and B, denoted by AB is only

possible when n = k . The resultant matrix will

have the dimensionsm x l.

The product is taken as the sum of the products

of the elements in the rows of the first matrix and

the elements in the columns of the second

matrix. Matrix multiplication in non commutative; that

is AB BA.



68/75

EXAMPLE 8.2:

a. Find DE and ED.b. Find the product of F and G.

Given:

E = [1 1; 2 1] D = [2 1; 1 1]

F = [ 1 0 4; 2 1 1; 3 1 0; 0 2 2 ]G = [ 2 4; 1 1; 3 0 ]



69/75

IDENTITY MATRIX

An identity matrix is a square matrix that

when multiplied to a non zero matrix, results

in the matrix itself.

A . I = A

An identity matrix has its diagonal elements

as all ones, with all the other elements zero.



70/75

TRANSPOSE OF A MATRIX

The transpose of a matrix A, denoted as A; is

the original matrix A with its rows and

columns interchanged.

A matrix that does not change its rows and

columns when transposed is called a

symmetric matrix.



71/75

ZERO ONE MATRICES

Matrices whose elements are either a

zero or a one is called a zero -one matrix.

Zero one matrices are important inrepresenting discrete structures.



72/75

JOIN & MEET OF ZERO ONE MATRICES

The join and meet of two zero one

matrix is analogous to addition and

multiplication of matrices.

However, instead of taking the sum, we

take the Boolean operation ofOR (join)

and AND (meet) on the matrices.


BOOLEAN PRODUCT OF ZERO ONE


73/75

BOOLEAN PRODUCT OF ZERO ONE

MATRICES

As with the join and meet, the Boolean

Product is analogous with the multiplication of

Matrices.

However, addition is replaced with OR and

multiplication with AND.

The Boolean Product of Zero One Matrices

A and B is denoted by A B.



74/75


75/75

ece math 311_topic 5 to 9

Documents