A set is a collection of distinct objects. Sets can be written using curly braces and commas to separate elements. An element belongs to a set if it is a member. The number of elements in a set is its cardinality. There are different methods to name sets such as roster and rule methods. A set is a subset of another set if all its elements are also elements of the other set. There are several operations that can be performed on sets such as union, intersection, difference, and complement.

1. Chapter 2.2
SETS

2. A set is a well-defined collection of objects such
as set of letters, set of numbers, set of books,
set of students, set of stars, sala set, etc.
A set is written using the following conventions:
a) Use capital English letters to denote sets;
b) Enclose elements using curly braces; and
c) Use commas to separate elements.

4. An element, denoted by ∈ (Greek letter Epsilon), is
any object that belongs to, or a member of a set.
The set of students may be written as
P = {Allan, Bj, Clyde, Jovi, Wivven, Derick, George,
Khing}
Example:
Consider the following expressions.
Allan ∈ P, John ∉ P

5. Cardinality of a Set
The cardinality of set A is the number of
elements in set A. In symbol, the cardinality of
set A is written as n(A).
Example:
Set P has only eight elements. Therefore,
n(P) = 8.

6. Methods of Naming Sets
There are two common methods of naming sets. Each
method has its own advantage. Here are the two method.
Roster method is done by listing all the elements. This
method is also called tabular method.
Rule method is done by describing what the set is about. It
uses a variable to represent the elements of the set
accompanied by a descriptive phrase.

7. Roster method
M = { Monday, Tuesday,
Wednesday, Thursday, Friday,
Saturday, Sunday}
E = {2}
N = {1, 2, 3, 4, …}
Rule method
M = { x ∣ x is a day in a week}
“M is the set of all x such that x
is a day in a week”
E = {x ∣ x is an even prime
number}
“E is the set of all x such that x
is an even prime number”
N = {x ∣ x is the set of counting
numbers}

8. The Universal Set
The universal set, denoted by U, is the set that
contains all the elements under consideration. It is
dependent on the limit put for its coverage. That is,
the set of residents of La Trinidad is the universal set
if it is the limit of discourse. The set of faculty
members of PSU, the set of flowers along the Agno
River, the set of gasoline stations in Pangasinan, etc.
are examples of a universal set provided all
discussions revolve only on the elements of such
sets.

9. SUBSETS
Set A is a subset of set B, denoted by A ⊆ B , if all
elements of A are also elements of B. Set A is a
“proper” subset of set B denoted by A ⊂ B , if all
elements of set A are also in set B but some elements
of set B are not in set A. For instance, the set of all
freshman students of PSU is a subset of the set of all
students in that University.

10. Types of Sets
Finite vs. Infinite Set.
A finite set is a set with elements that are countable
one by one. Its cardinality can be assigned a certain
number.
An infinite set on the other hand is a set that is not
finite. In other words, it is a set with elements that are
not countable one by one.

11. Types of Sets
Equal vs. Equivalent Sets.
Set A is equal to B, denoted by A = B , if they have
exactly the same
elements.
Example
The sets A = {e, i, o, u, a} and B = {a, e, i, o, u} are
equal because both of these sets contain all the
vowels of the English alphabet.

12. Set A is equivalent to set B, denoted by A ~ B, if they
have the same cardinality. This means that sets with
the same number of elements, regardless of the
nature of the elements are equivalent.
Example:
The sets A= {e, i, o, u, a} and C = {1, 2, 3, 4, 5} is a
pair of equivalent sets because the cardinality of both
of them is 5.

13. Types of Sets
Unit Set vs. Null Set.
A unit set is a set that contains only one element. The
set of capitals of a province is an example of a unit set
since each province has only one capital.
An empty set, or a null set is a set that contains no
elements. The Greek letter phi, ∅, is used to symbolize
the null set. An empty curly brace { } may also be used
to denote an empty set.

15. Operations on Sets
1. Set A union set B, denoted by A ∪ B , is the set of all
elements that belong to set A, or set B, or to both
set A and set B.
2. Set A intersection Set B, denoted by A ∩ B , is the set
of all elements that belong to both set A and set B.

16. 3. Set difference. The difference of two sets A and B,
denoted by A − B , is the set of all elements that are
in set A, but not in set B.
4. Complement of a set. The complement of set A,
denoted by A’ (or A-prime), is the set of all elements of
the universal set that are not in set A.

17. Example:
Consider the following sets.
U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
A = {0, 1, 2, 3, 4} B = {0, 2, 4, 6, 8} C = {1, 3, 5, 7, 9}
1. A ∪ B = {0, 1, 2, 3, 4, 6, 8}
2. A ∪ C = {0, 1, 2, 3, 4, 5, 7, 9}
3. A ∩ B = {0, 2, 4}
4. B ∩ C = { } = ∅

18. Example:
Consider the following sets.
U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
A = {0, 1, 2, 3, 4} B = {0, 2, 4, 6, 8} C = {1, 3, 5, 7, 9}
5. A – B = {1, 3}
6. A – C = {0, 2, 4}
7. A’ = {5, 6, 7, 8, 9}
8. B’ = {1, 3, 5, 7, 9}
9. C’ = {0, 2, 4, 6, 8}

19. Application of Sets
Sets and other related concepts can be used to solve real life
problems such as the following:
Example:
Ma’am Xinom and Ma’am August are very generous teachers. On a Friday
ma’am Xinom and ma’am August, decided to treat their math 21 students to a
serving of French fries each with the following choices of dips: catsup,
mayonnaise, and mustard. Among their 42 students, 26 wanted mustard as a dip,
19 wanted mayonnaise, 17 wanted catsup, 11 wanted mustard and mayonnaise,
10 wanted mustard and catsup, 7 wanted mayonnaise and catsup and 6 wanted all
three as dips. How many students wanted (a) catsup dip only; (b) mustard but not
mayonnaise; (c) mayonnaise but not mustard; (d) catsup and mustard but not
mayonnaise; (e) mayonnaise and mustard but not catsup; and (f) none of the three
dips?

20. Given the following sets:
U = {g, h, i, j, k, l, m, n, o}
A = {g, h, i, j, k} B = {j, k, l, m}
C = {k, l, m, n, o} D = {g, i, k, m, o} E = {h, j, l, n}
1. A ∩ B
2. B ∪ D
3. (A – E)’
4. (A ∪ B) – (C ∪ D)
5. C ∩ D’