University of La Salette, Incorporated
College of Engineering and Architecture
PCEA 020 - Advanced Engineering Algebra
Module 1 - Introduction to Advanced Engineering Algebra and Set Theory
Pre-Assessment
Answer what is being asked in each question in a ¼ sheet of paper.
1. Who is the father of Algebra?
2. Give 3 practical applications of Algebra to Engineering.
3. Give 3 attributes of a student must have to ace Advanced Engineering Algebra.
Objectives
At the end of this module, students are expected to:
 Understand the basic principles, key elements, and practical applications of algebra
within engineering.
 Discover the complexities of algebraic equations in relation to the field, while
developing your problem-solving skills through the exploration of techniques and
strategies.
 Apply and analyse the use of modern algebra to several engineering worded problems.
Introduction
One might be wondering what Algebra Engineering is all about. This branch of engineering
involves the application of algebraic techniques to solve complex engineering problems. It acts as the
bridge between abstract mathematical concepts and practical engineering solutions. Delve deep into the
fascinating world of Algebra Engineering with this comprehensive course. With detailed introductions to
key concepts and illustrated case studies, you'll gain a thorough understanding of the integral role of
algebra in engineering. Algebra helps to represent physically occurring phenomena in mathematical
terms. This enables engineers to make mathematical predictions about their designs and systems and
then check these predictions with the actual performance. Algebra in Engineering can be defined as the
use of algebraic equations to model, analyse and design complex engineering systems such as circuits
and infrastructure.
Definition
Algebra (from an Arab word “al-jabr” which means reunion of broken
parts) studies symbols and the rules of manipulating them. Algebra (Al-jabr) as a
subject itself comes from the title of the early 9th century book of Muhammad
ibn Musa al-Khwarizmi, the Ilm al-jabr wa l-muqabala.
Muhammad ibn Musa al-Khwarizmi was a Persian mathematician who
was partly influenced by his Hindus and Greeks predecessors.
Basic Principles of Engineering Algebra
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Equation formulation: Algebra Engineering first involves forming equations that
represent an engineering system or problem.
Solution of equations: Once the equations are formed, various methods can be used to
solve these algebraically.
Application of solutions: The solutions derived are then applied in the engineering
problem or system.
Common Terminologies in Algebra
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Variables and Constants - variables are quantities that can change, whereas constants are
quantities that remain unchanged in an algebraic equation.
Equation – a mathematical representation of a system that is equal in nature.
Prepared by: Engr. Richard V. Avellanoza, CE, SO, RMP
University of La Salette, Incorporated
College of Engineering and Architecture
PCEA 020 - Advanced Engineering Algebra
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Algebraic Equation – an equation that is represented by numbers and letters.
Solution – a detailed set of calculation or computation that arrives to the best answer to a
problem.
Terms – can be an independent or dependent mathematical representation that may and may
not have a value or significance (ex. 3X, 7, 4C⁴)
Like Terms – are terms that have the same degree. It can be combined through addition or
subtraction. (ex. 6X and 3X are like terms)
Numbers – an entity describing the quantity or position of a mathematical object or extension of
these concepts.
Numerals- figures, signs, symbols or group of these denoting a number.
Set Theory
SETS – a collection well defined objects; the objects are called members of set or elements of the set.
U
A
All points inside Circle A are elements of Set A. (Collection of things, objects, or elements that are
contained in the group)
NOTATIONS
Tabular Form – this form can be done by listing its elements, normally done in a set with definite
number of elements.
For example:
A = {2,3,4,5,6,7}, to be read as “A is the set whose elements are the numbers 2,3,4,5,6,7.”
Set Builder Form – This form can be done by defining the properties of the elements of the set, for
definite and indefinite number of elements.
For example:
A = {x|x is an integer, x > 5}, to be read as “A is the set of x such that x is an integer and x is greater than
5.”
“p is an element of A”, can be written as “p A”
“p is not an element of A” can be written as “p A”
Common notations for the set of numbers are the following:
N = set of non-negative integers (0,1,2,3…)
P = set of positive integers (1,2,3…)
Z = set of integers (…, -1,-2,0,1,2)
Q = set of rational numbers
R = set of real numbers
C = set of complex numbers
Thus, A = {x|x is an integer, x > 5} is written also as A = {x|x
Z, x > 5}
UNIVERSAL SET – A set that contains all possible elements and is usually denoted as U.
U
Prepared by: Engr. Richard V. Avellanoza, CE, SO, RMP
University of La Salette, Incorporated
College of Engineering and Architecture
PCEA 020 - Advanced Engineering Algebra
All points contained in the rectangle are elements of the universal set.
SUBSET – A set such that all of its elements are just part of a larger set. Such as, all elements of set B are
also elements of set A, hence, can be denoted as “B A”
U
A
B
All points that belong to “B” belong to “A”.
PROPER SUBSET – A proper subset of A is a subset of A that is not equal to A. In other words, if B is a
proper subset of A, then all elements of B are in A.
For example:
If A = {1,3,5}, then B = {1,5} is a proper subset of A.
The set C = {1,3,5} is a subset A, but is not a proper subset of A since C = A.
COMPLEMENT OF A SET – It refers to elements that are not members of a certain set, but element of
universal set. The complement of set A can be written in the form A^c or A’.
U
A
A’
NULL SET or EMPTY SET – ( ) – any set that does not contain any element.
VENN DIAGRAM – a visual representation of sets
RELATIVE COMPLEMENT – the relative complement of B with respect to A, denoted by A\B, is the set of
elements which belong to A but do not belong to B.
UNION OF SETS – The union of two sets A and B, denoted by A U B, is the set of all elements that belong
to A or B. We can say that an element is a part of union as long as it is a member of at least one of the
sets (elements need not be member of both A and B).
Example: Determine the elements that belong to A U B.
Prepared by: Engr. Richard V. Avellanoza, CE, SO, RMP
University of La Salette, Incorporated
College of Engineering and Architecture
PCEA 020 - Advanced Engineering Algebra
A U B = {1,2,3,4,5,6,7,8,9,10,11,12}
INTERSECTION OF SETS – the intersection of two sets A and B, denoted by A B, is the set of all
elements which belong to A and B. Elements that belongs to A and B should be member of A and at the
same time member of B.
Example: Determine the elements that belongs to A
A
B
B = {6,7}
PRINCIPLE OF INCLUSION AND EXCLUSION
It provides an organized method to find the number of elements in the union of a given group of
sets, the size of each set, and the size of all possible intersections among the sets.
It is a counting technique that includes all elements in the set/s appearing not more than once.
If such element has occurred (e.g counted twice), the subsequent repeating element is not counted.
For example, let say you have two sets A and B, the equation becomes,
A U B = (A + B) – (A
B)
So, if you have three sets, A, B, and C, the equation becomes
Furthermore, if you have four sets say A,B,C and D.
Prepared by: Engr. Richard V. Avellanoza, CE, SO, RMP
University of La Salette, Incorporated
College of Engineering and Architecture
PCEA 020 - Advanced Engineering Algebra
References:
(2023). Advanced Engineering Algebra
https://www.studysmarter.co.uk/explanations/engineering/engineering-mathematics/algebraengineering/#:~:text=Algebra%20Engineering%20involves%20the%20use,and%20construction%20of%2
0algebraic%20bridges.
(2023). Real Numbers and Properties of Numbers
https://www.storyofmathematics.com/absolute-value-inequalities/
Prepared by: Engr. Richard V. Avellanoza, CE, SO, RMP