IMEMATH Notes
MATRIX a rectangular array of numbers.
a11
a
A 21
a m1
a12 a1n
a 22 a 2 n
a m 2 a mn
Where: aij = elements found in the ith row, jth column
i = row
j = column
m = no. of rows
n = no. of columns
VECTOR an ordered list of n numbers
Ro
IMEMATH Notes
Systems of Linear Equations
Conditions: No. of Equations = No. of unknowns or unique
solutions
Consider a linear system of n equations in n unknowns:
Where aij and bj are constants and X1, X2, X3,.,Xn are
unknowns for i=1,2,3,.,n and j=1,
OPTIMIZATION
IMEMATH Notes
Basic Concepts
Optimization is the state of either maximizing or
minimizing a chosen measure of effectiveness
subject to certain limitations or constraints.
Some Common Measures of Effectiveness
1)
2)
3)
4)
5)
6)
Profit
Cost
V
Introduction to Logic
IMEMATH Notes
History
The first systematic study of logical reasoning is found in the
work of Greek philosopher Aristotle (384 - 322 BC). In his
treatises on logic, he presented a collection of principles for
deductive reasoning. The
SIMULTANEOUS
NON-LINEAR EQUATIONS
IMEMATH Notes
Non-Linear Equations
A non-linear system can be represented by:
The vector x* = [x1*, x2*, xn*] is a solution to
P1 = x* satisfies f1(x) i.
Methods for Solving Simultaneous
Non-Linear Equations
Newton-Rap
Methods of Proof, Mathematical
Induction, Predicates and
Quantifiers
IMEMATH Notes
Methods of Proof
In all mathematical proofs, there is a
collection of statements called
hypotheses and a statement called the
conclusion.
The conclusion must be proven to
f
SYSTEMS OF LINEAR EQUATIONS
IMEMATH Notes
Systems of Linear Equations
Conditions: No. of Equations = No. of unknowns or
unique solutions
Consider a linear system of n equations in n
unknowns:
Where aij and bj are constants and X1, X2, X3,.,Xn are
unknowns
SOLUTION OF NONLINEAR EQUATIONS
IMEMATH Notes
ROOT FINDING
Given:
f(x) = 0
Where: f(x) is a simple non-linear
equation in one variable
Y = f(x)
y-axis
x1
Roots
x2
x- axis
x3
ROOT FINDING
A
function of the form f(x) = 0
can be converted to the form x =
g(
FUNDAMENTALS OF LOGIC
EXERCISES
A. Determine whether each of the following sentences is a statement:
1) In 1990 George Bush was the president of the United States.
2) x + 3 is a positive integer.
3) If only every morning could be as sunny and clear as thi
Linear Algebra
IMEMATH Notes
Matrix and Its Properties
MATRIX a rectangular array of numbers.
a11
a
A 21
a m1
a12 a1n
a 22 a 2 n
a m 2 a mn
Where: aij = elements found in the ith row, jth column
i = row
j = column
m = no. of rows
n = no. of colum
INDUSTRIAL ENGINEERING MATHEMATICS
Problems in Optimization Techniques
1. An industrial engineer wants to design a silo, which consists of cylinder
surmounted by a hemisphere. The said silo is to carry a fixed capacity V. The
floor is twice as expensive a
SET THEORY
EXERCISES
A. Determine the set builder notation for the following:
1) D F
2) DF
3) F D
4) D F D F
5) F D E
B. Determine which of the following statements are true and which are false.
1) Z Q
6) Z R R
2)
3)
4)
5)
Z
Q
R
Q
Q
R
Q
R Q
7)
8)
10
Methods of Proof,
Mathematical Induction,
Predicates and Quantifiers
IMEMATH Notes
Methods of Proof
In all mathematical proofs, there is
a collection of statements called
hypotheses and a statement called
the conclusion.
The conclusion must be proven to
f
SIMULTANEOUS
NON-LINEAR EQUATIONS
IMEMATH Notes
Non-Linear Equations
A non-linear system can be represented
by:
The vector x* = [x1*, x2*, xn*] is a
solution to P1 = x* satisfies f1(x) i.
Methods for Solving Simultaneous
Non-Linear Equations
Newton-Rap
OPTIMIZATION
IMEMATH Notes
Basic Concepts
Optimization is the state of either maximizing
or minimizing a chosen measure of
effectiveness subject to certain limitations or
constraints.
Some Common Measures of Effectiveness
1)
2)
3)
4)
5)
6)
Profit
Cost
V
Introduction to Logic
IMEMATH Notes
History
The first systematic study of logical reasoning is found
in the work of Greek philosopher Aristotle (384 - 322
BC). In his treatises on logic, he presented a collection
of principles for deductive reasoning. The