- Information
- AI Chat
Module 1 pdf - A practice material for math lovers and engineers
Civil Engineering (CE 001)
Eastern Visayas State University
Recommended for you
Preview text
Module 1: Real Number System
Learning Outcomes:
At the end of this lesson, students are expected to:
Describe a set by roster or rule method
Perform operations on sets
Classify numbers as natural, integer, rational or irrational
Apply the properties of real numbers to specific examples.
I. THE REAL NUMBER SYSTEM
The real number system evolved over time by expanding the notion of what we mean by the
word “number.” At first, “number” meant something you could count, like how many sheep a
farmer owns. These are called the natural numbers, or sometimes the counting numbers.
Natural Numbers or Counting Numbers (N) - these are the numbers we use in counting
things.
1, 2, 3, 4, 5,...
The three dots, called ellipsis, indicate that the pattern continues indefinitely.
At some point, the idea of “zero” came to be considered as a number. If the farmer does not
have any sheep, then the number of sheep that the farmer owns is zero. We call the set of
natural numbers plus the number zero the whole numbers.
Whole Numbers (W) - Natural numbers and zero
0, 1, 2, 3, 4, 5,...
Integers (Z) - counting numbers, their opposites and zero.
...-3, - 2, - 1, 0, 1, 2, 3,...
Rational Numbers (Q) – any number that can be written as a fraction. It includes repeating
decimals that terminate (end).
− 3
1
,
2
3
, 0,
− 5
7
, - 1,
3
8
Irrational Numbers (I) – a real number that is not rational. Decimal that does not terminate or
repeat.
√ 2 , √ 5 , √ 11 , 𝜋
The Real Numbers (R)
Rationals + Irrationals
All points on the number line
Or all possible distances on the number line
When we put the irrational numbers together with the rational numbers, we finally have the
complete set of real numbers. Any number that represents an amount of something, such as a
weight, a volume, or the distance between two points, will always be a real number. The
following diagram illustrates the relationships of the sets that make up the real numbers.
For further explanation you may watch videos here:
youtube/watch?v=0OwvN-957aE – The Real Number System (6:21 minutes)
youtube/watch?v=2gzt5izv5Ds - Classifying Numbers - Integers, Whole &
Natural Numbers, Real & Imaginary Numbers (10:08 minutes)
Worksheet 1: Refer to classwork in google classroom.
II. SETS
Look around your room! Are things properly arranged? Do you think you can easily tell where
your books are? What about your shirts? Your shoes? If you answered “yes”, may I asked how
did you arrange your stuffs? _______________________________
“𝑏𝑜𝑜𝑘 is an element in set 𝐴”. In symbol, 𝑏𝑜𝑜𝑘 ∈ 𝐴.
“𝑚𝑒𝑙𝑜𝑛 is an element in set 𝐵”. In symbol, 𝑚𝑒𝑙𝑜𝑛 ∈ 𝐵.
“𝑝𝑎𝑟𝑎𝑐𝑒𝑡𝑎𝑚𝑜𝑙 is an element in set 𝐶”. In symbol, 𝑝𝑎𝑟𝑎𝑐𝑒𝑡𝑎𝑚𝑜𝑙 ∈ 𝐶.
“𝑔𝑟𝑜𝑢𝑛𝑑 𝑝𝑜𝑟𝑘 is an element in set 𝐷”. In symbol, 𝑔𝑟𝑜𝑢𝑛𝑑 𝑝𝑜𝑟𝑘 ∈ 𝐷.
On the other hand, we can say that:
“𝑐ℎ𝑖𝑐𝑘𝑒𝑛 is not an element in set 𝐵”. In symbol, 𝑐ℎ𝑖𝑐𝑘𝑒𝑛 ∉ 𝐴.
“𝑜𝑟𝑎𝑛𝑔𝑒𝑠 is not an element in set 𝐶”. In symbol, 𝑜𝑟𝑎𝑛𝑔𝑒𝑠 ∉ 𝐶.
Writing a Set
There are two methods of writing a set: (1) the roster method also known as the list
method, and (2) the set-builder notation also known as the rule method.
The roster method is defined as a way to show the elements of a set by listing the elements
inside of brackets.
Sets 𝐴, 𝐵, and 𝐶 above, are all described in a roster method because all the elements are written
inside the set. Set 𝐴 has all its elements listed in the set, namely: 𝑏𝑜𝑜𝑘, 𝑛𝑜𝑡𝑒𝑏𝑜𝑜𝑘, 𝑏𝑎𝑙𝑙 𝑝𝑒𝑛,
𝑝𝑒𝑛𝑐𝑖𝑙. Set 𝐵 also showed that only the following member fruits are included in the set:
𝑎𝑝𝑝𝑙𝑒, 𝑏𝑎𝑛𝑎𝑛𝑎, 𝑜𝑟𝑎𝑛𝑔𝑒, 𝑚𝑒𝑙𝑜𝑛, and 𝑔𝑟𝑎𝑝𝑒𝑠.
The rule method, a property of a set which defines whether an object is an element of the set or
not is specified and enclosed in braces.
If the rule method is used to describe the sets in examples 1 and 2, we have the following:
A = {x | x is a positive odd integer less than 15}
This read as “A is the set of all x’s such that x is positive odd integer less than 15.
B = {x | x is a counting number}
This read as “B is the set of all x’s such that x is a counting number.”
For further explanation you may watch videos here:
youtube/watch?v=yWidD7lBsv8 - Introduction to Sets Part 1 - Roster & Rule
Method I Señor Pablo TV (11:22 minutes)
youtube/watch?v=_W70arQDpsc - Introduction to Sets - Lesson 1(Roster and
Set-builder Method) (9:28 minutes)
Worksheet 2 : Refer to classwork in google classroom.
Operations on Sets
The new set formed if you combine all the elements in two sets is called the union
of the sets, denoted by the symbol ∪. Remember that elements which are found in the
two sets should be written only once.
The set of all the elements common between two sets is called the intersection
of two sets and denoted by the symbol, ∩. Two sets which has no common element
or whose intersection is a null set are called disjoint sets.
Example. Given the following sets: 𝐴 = { 2 , 4 , 6 , 8 , 10 }, 𝐵 = { 1 , 3 , 5 , 7 , 9 }, 𝐶 =
{
2 , 3 , 5 , 7
}
.
Find: (a) 𝐴 ∪ 𝐵; (b) 𝐵 ∪ 𝐶; (c) 𝐴 ∪ 𝐶 ; (d) 𝐴 ∩ 𝐵; (e) 𝐵 ∩ 𝐶
Answers:
(a) 𝐴 ∪ 𝐵 =
{
2 , 4 , 6 , 8 , 10
}
∪
{
1 , 3 , 5 , 7 , 9
}
=
{
1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10
}
(b) 𝐵 ∪ 𝐶 = { 1 , 3 , 5 , 7 , 9 } ∪
{
2 , 3 , 5 , 7
}
=
{
1 , 2 , 3 , 5 , 7 , 9
}
(c) 𝐴 ∪ 𝐶 = { 2 , 4 , 6 , 8 , 10 } ∪
{
2 , 3 , 5 , 7
}
=
{
2 , 3 , 4 , 5 , 6 , 7 , 8 , 10
}
(d) 𝐴 ∩ 𝐵 = { 2 , 4 , 6 , 8 , 10 } ∩ { 1 , 3 , 5 , 7 , 9 } = { }
(e) 𝐵 ∩ 𝐶 =
{
1 , 3 , 5 , 7 , 9
}
∩
{
2 , 3 , 5 , 7
}
=
{
3 , 5 , 7
}
Look at the intersection of sets 𝐴 and 𝐵. It is a null set because they do not have
any common element, hence, they are called disjoint sets.
Can we subtract two sets? Yes, of course. What is new in subtraction of two sets
is that the elements in your second set may not necessarily be all found in the first set.
The difference of two sets 𝐴 and 𝐵 denoted by 𝐴 − 𝐵, is the set of elements in 𝐴 which
do not belong to 𝐵.
Example: Given the following sets: 𝐴 =
{
2 , 4 , 6 , 8 , 10
}
, 𝐵 =
{
1 , 3 , 5 , 7 , 9
}
, 𝐶 =
{
2 , 3 , 5 , 7
}
.
Find: (a) 𝐴 − 𝐵; (b) 𝐵 − 𝐴 ; (c) 𝐵 − 𝐶 ; (d) 𝐶 − 𝐵 ; (e) 𝐶 − 𝐴
(a) 𝐴
′
= 𝑈 − 𝐴 =
{
1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19
}
−
{
4 , 8 , 12 , 16
}
𝐴
′
=
{
4 , 8 , 12 , 16
}
(b) 𝐶
′
= 𝑈 − 𝐶
𝐶′ =
{
1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19
}
−
{
2 , 3 , 5 , 7 , 11 , 13 , 17 , 19
}
𝐶
′
=
{
1 , 4 , 6 , 8 , 9 , 10 , 12 , 14 , 15 , 16 , 18
}
(c) 𝐴 − 𝐵 = { 4 , 8 , 12 , 16 } − { 3 , 6 , 9 , 12 , 15 , 18 } = { 4 , 8 , 16 }
(d) 𝐷 ∪ 𝐸 =
{
6 , 12 , 18
}
∪
{
5 , 10 , 15
}
=
{
5 , 6 , 10 , 12 , 15 , 18
}
(e) 𝐴 ∩ 𝐷 =
{
4 , 8 , 12 , 16
}
∩
{
6 , 12 , 18
}
=
{
12
}
Another operation on sets that has a great importance in higher mathematics is to
find the set product (also known as Cartesian product or cross product) of two sets.
To find the cross product of sets 𝐴 and 𝐵, denoted by 𝐴 × 𝐵(read “A cross B”), is the set
of all possible ordered pairs (𝑎, 𝑏) where 𝑎 ∈ 𝐴 and 𝑏 ∈ 𝐵.
Example: Given the sets 𝐴 = { 1 , 2 } and 𝐵 = {𝑎, 𝑏, 𝑐}. Find: (a) × 𝐵 ; and (b) 𝐵 × 𝐴.
Answers:
𝐴 × 𝐵 = {( 1 , 𝑎), ( 1 , 𝑏), ( 1 , 𝑐), ( 2 , 𝑎), ( 2 , 𝑏), ( 2 , 𝑐)}
𝐵 × 𝐴 = {(𝑎, 1 ), (𝑎, 2 ), (𝑏, 1 ), (𝑏, 2 ), (𝑐, 1 ), (𝑐, 2 )}
Remember that in any ordered pairs,
(
𝑎, 𝑏
)
≠
(
𝑏, 𝑎
)
.
A very useful method of illustrating relations and operations involving sets is the
Venn Diagram (Venn-Euler Diagram). A rectangle is used to represent the universal
set and circles represent subsets. Overlapping circles shows sets having common
description while circles that do not overlap do not have common trait.
A Venn diagram may help you come up with a decision if you are torn between
options. Look at the diagram below. It shows the features of the two boarding houses
or apartment that you can choose from. The diagram tells you that both has TV,
refrigerator, kitchenet and Wi-Fi. Apartment 𝐴 has 1 double bed, and hot and cold water
supply. While apartment 𝐵, has stove top, swimming pool, 1 double deck bed and
parking.
This diagram is giving you a clearer picture of what they offer and choose the
most convenient apartment that best fit to your need.
Example: Let the universal set represents the set of the goods available in all the
stores in the nearby area.
Store 𝐴 sells ball pens, beauty soap, canned foods, coffee, detergents, medicines,
milk, papers, and pentel pens.
Store 𝐵 sells banana cue, canned foods, coffee, detergents, eggs, fried chicken,
milk, noodles, papers, pencils, sandwiches, and toothpastes.
Store 𝐶 sells bags, ball pens, books, masking tapes papers, pencils, pentel pens,
and staplers.
Write the set of items sold in each store in a roster method.
Create a Venn diagram of the given sets. Describe the sets obtained in the
following operations: (a) 𝐴 ∪ 𝐵 ∪ 𝐶; (b) 𝐵 ∩ 𝐴; (c) 𝐵 ∩ 𝐶; (d) 𝐴 ∩ 𝐵 ∩ 𝐶;
(e) 𝐴 − 𝐵; (f) 𝐶 − 𝐴 ; (g) 𝐶′ ;
Answers:
𝐴 = { ball pens, beauty soap, canned foods, coffee, detergents,
medicines, milk, papers, and pentel pens}
(c) 𝐵 ∩ 𝐶 = { papers, pencils}
Papers and pencils are sold in stores 𝐵 and 𝐶.
(d) 𝐴 ∩ 𝐵 ∩ 𝐶 =
(
𝐴 ∩ 𝐵
)
∩ 𝐶
={canned foods, coffee, detergents, milk, papers} ∩ {bags, ball pens,
books, masking tapes papers, pencils, pentel pens, and staplers}
𝐴 ∩ 𝐵 ∩ 𝐶 = {papers}
This set shows that papers are available in all the three stores.
(e) 𝐴 − 𝐵 = { ball pens, beauty soap, medicines, pentel pens}
This set shows that items available in store 𝐴 which are not found in
store 𝐵.
(f) 𝐶 − 𝐴 = { bags, books, masking tapes, pencils, staplers}
This set shows that items available in store 𝐶 which are not found in
store 𝐴.
(g) 𝐶
′
= {banana cue, beauty soap, canned foods, coffee, detergents, eggs,
fried chicken, medicines, milk, noodles, sandwiches, and toothpastes,}
The set shows the items which are sold in the nearby stores but are not
sold in store C.
For further explanation you may watch this video:
youtube/watch?v=6sEd_gAdKsI - Set Operations || Mathematics in the Modern
World (18:57 minutes)
Worksheet 3 : Refer to classwork in google classroom.
Module 1 pdf - A practice material for math lovers and engineers
Course: Civil Engineering (CE 001)
University: Eastern Visayas State University
- Discover more from:
