This document provides an overview of sets, elements, and cardinality. It defines a set as a collection of well-defined objects called elements. Sets are represented by capital letters and elements belong to a set if they are included in it. The number of elements in a set is its cardinality. Examples demonstrate sets, elements, and properties like subsets and the universal set. Key points reiterate the definitions and properties of sets. An activity at the end asks students to write sets in roster form.
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Lecture 01 Sets.pdf
1. LESSON 01:
SETS
Engr. Onofre E. Algara, Jr.
BS Electrical Engineering (DLSU-D)
MS Electrical Engineering (Mapua University)
Course Instructor
COLEGIO DE MUNTINLUPA
ELECTRICAL ENGINEERING DEPARTMENT
2. SETS, ELEMENTS AND CARDINALITY
• A set is a collection of well-defined and distinct objects.
• These objects are usually referred to as the elements of that
set.
• Sets are denoted by uppercase letters of the alphabet such as
A, B , C, S and T.
• α is an element of set A iff α belongs to the set A. In symbols,
α ∈ 𝐴.
• b is not an element of set A iff b does not belongs to the set A.
In symbols, b ∉ 𝐴.
• The number of elements of a set is called the cardinality of
that set. The cardinality of set A is denoted by |A|.
3. EXAMPLES
1. Set A containing the subjects that Grade 7 students will
study this school year is a well-defined set.
•All Grade 7 students will study the same set of subjects.
•Mathematics ∉ 𝐴 because it is one of the subjects that
Grade 7 students will study this school year.
4. EXAMPLES
2. Set Set B containing the difficult subjects that Grade 7
students will study this school year is not a well- defined
set.
•Some of the students may have difficulty in mathematics,
while other students may find mathematics easy.
•Thus, it could not be determined whether or not
mathematics is an element of set B.
5. UNIVERSAL AND EMPTY SET
•The universal set is the set containing all elements being
considered or studied. It is usually denoted by U.
•An empty set, denoted by { } or ∅, is a set that does not
contain any element. It is also known as the null set.
6. EXAMPLES
•Two sets, M and N, form the universal set.
•Set M contains the letters in the word "examples" and set N
contains the even numbers greater than four but less than
eight.
1. The universal set is {e, x, a, m, p, l, s, 6}.
2. From the universal set, the set containing odd numbers
is { } because the universal set does not contain any odd
number.
7. SUBSET
•A set is a subset of another set if all elements of the first
set are also elements of the second set. The symbol ⊆ is
used to denote subset.
•A set can have 2𝑛 subsets, where n is the number of
elements of the considered set. A subset may contain no
element, one element, two elements, or even all the
elements of the original set.
8. EXAMPLE
Given set B {x, y, z}, find the following:
1. elements of set B
2. cardinality of set B
3. all the subsets of set B
• The elements of set B are x, y, and z. This answer can also be
written using the element symbol: x ∈ B, y ∈ B, and z ∈ B.
• Set B has three elements. Thus, the cardinality of set B is 3 or
|B| = 3.
• Since set B has three elements, it has 23 or 8 subsets. These
are ∅, {x}, {y}, {z}, {x, y}, {y, z}, {x, z}, and {x, y, z}.
9. KEY POINTS
•A set is a collection of well-defined and distinct objects.
•The objects in a set are called its elements.
•A set with no elements in it is called an empty set denoted
by the symbols { } or ∅.
•The cardinality of a set is the number of elements in the
set.
•The universal set is the set containing all elements being
studied.
•A set is a subset of another set if all elements of the first
set are also elements of the second set.
10. SEATWORK
• Write the following sets in the
roster form.
(a) A = The set of all even numbers
less than 12
(b) B = The set of all prime numbers
greater than 1 but less than 29
(c) C = The set of integers lying
between -2 and 2
(d) D = The set of letters in the
word LOYAL
(e) E = The set of vowels in the
word CHOICE
(f) F = The set of all factors of 36
(g) G = {x : x ∈ N, 5 < x < 12}
(h) H = {x : x is a multiple of 3 and x < 21}
(i) I = {x : x is perfect cube 27 < x < 216}
(j) J = {x : x = 5n - 3, n < 3}
(k) M = {x : x is a positive integer and x2 < 40}
(l) N = {x : x is a positive integer and is a
divisor of 18}
(m) P = {x : x is an integer and x + 1 = 1}
(n) Q = {x : x is a color in the rainbow}
