THE LANGUAGE OF SET
MATHEMATICAL LANGUAGE
Core Idea:​
Like any language,
mathematics has its own
symbols, syntax and rules.​
Importance of the Language of Mathematics:
•To understand the expressed ideas
•To communicate ideas to others
Four Basic Mathematical Concepts:
Sets, Relations, Functions, Binary Operations
THE LANGUAGE OF SET
LEARNING OUTCOMES
At the end of this lesson, you should
be able to:
 Discuss the concept of
sets.
 Represent sets using
roster method and rule
method.
SETS
SET – a well-defined
collection of
distinct objects.
Definition and
Facts about Sets
1. well-defined: there is
no ambiguity in
deciding whether or not
a given object belongs
to a set
2. distinct: the objects
must be distinguishable
from each other.
Example # 1:​
A collection of students in
an English class is a set.
Example # 2:​
A collection of all the days
of the week is a set.
A collection of ten most talented
women in the Philippines
NOT a set
A collection of five best
basketball players of all time
NOT a set
 Sets are denoted by capital
letters (e.g. A, D, C,…). The
objects of a set are separated
by comma and are enclosed
by braces.
 These objects are called
Definition and
Facts about Sets
the elements or members of
the set.
 If A is a set, the notation x ∈ A
means that x is an element of
A. (x belongs to A)
 The notation x ∉ A means that
x is not an element of A.
N is the set of
positive integers
less than 10
 A set contains the months of the
year beginning with the letter J
M = {January, June, July}
How do we
write a set and
its elements?
January ∈ M
August ∉ M
 A set contains the first five
composite numbers
C= {4, 6, 8, 9, 10}​
8∈C
14 ∉ C
1. ROSTER METHOD or
Listing Method
Ways of
describing sets
- list all the elements of
the set enclosed with
braces
Examples:​
M = {January, June, July}​
C= {4, 6, 8, 9, 10}
For sets with more elements,
Ways of
describing
sets
show the first few entries to
display a pattern, and use an
ellipsis to indicate “and so on.”
For example,
{1,2,3,…,50} represents the
set of the first 50 positive
integers.
Given
P = {0, {0}}​
Question:
1. What are the elements of P?
EXAMPLE
2. Are the elements the same?
Answer:
1. Elements of P are 0 and {0}
2. No. The symbol 0 represents
the number zero while {0}
represents a set that has one
element which is 0.
2. RULE METHOD or SET
BUILDER NOTATION
Ways of
describing sets
Let S denote a set and let P(x)
be a property that
elements of S may or may
not satisfy. We may define a
new set to be the set of all
elements x in S such that
P(x) is true.
{x ∈ S | P(x)}
The set of all
such that
Example # 1:​
a. The collection of all the
months of a year beginning
with the letter J
b. The collection of first five
composite numbers
EXAMPLES
Answers:
a. {x|x is a month of the year
which begins with letter J}
b. {y|y is the first five
composite number}
a. The collection of all the
months of a year beginning
with the letter J
b. The collection of first five
composite numbers
ROSTER METHOD:​
Roster Method
and Rule Method
M = {January, June, July}​
C= {4, 6, 8, 9, 10}
RULE METHOD:​
a. {x|x is a month of the
year which begins with
letter J}​
b. {y|y is the first five
composite number}
Example # 2:​
The set of natural
numbers greater than 4
EXAMPLE
Solution:
N - the set of natural
numbers
> - greater than
Answer:
{x| x ∈ N where x > 4} or​
{x| x ∈ N, x > 4} ​