CHAPTER 2: SPEAKING MATHEMATICALLY
SPEAKING MATHEMATICALLY
The aim of this chapter is to introduce you
to a mathematical way of thinking that can
serve you in a wide variety of situations.
Often when you start work on a mathematical
problem, you may have only a vague sense of how
to proceed. You may begin by looking at examples,
drawing pictures, laying around with notation, rereading the
problem to focus on more of its details, and so forth. The closer you get to a
solution, however, the more your thinking has to crystallize. And the more you
need to understand, the more you need language that expresses mathematical
ideas clearly, precisely, and unambiguously.
This chapter will introduce you to some of the special language that is a
foundation for much mathematical thought, the language variables, sets, relations
and functions. Think of the chapter like the exercises you would do before an
important sporting event. Its goal is to warm up your mental muscles so that you
can do your best.
This chapter will explain some concepts of statistics that are essential
for some quantitative researches. Specifically, this chapter will discuss the
following lessons:
 2.1 : Variables
 2.2 : The Language of Sets
 2.3 : Mathematical Symbols
This material is intended solely for the academic utilization of St. Francis Xavier College.
1
CHAPTER 2: SPEAKING MATHEMATICALLY
What is CHAPTER 2 about?
SPEAKING MATHEMATICALLY
LESSON
2.1
Variables
This lesson is designed for you to:
 explain what is a variable.
 rewrite sentences with the use of variables.
Interactive Discussion
What is a VARIABLE?
A variable is a quantity that may change within the context of a
mathematical problem or experiment. Typically, we use a single letter
to represent a variable. The letters x, y, and z are common generic
symbols used for variables. Sometimes, we will choose a letter that
reminds us of the quantity it represents, such as t for time, v for voltage,
or b for bacteria.
A variable is used as a placeholder when you want to talk about something by:
(1) you imagine that it has one or more values but you don’t know what they are;
(2) you want whatever you say about it to be equally true for all elements in a
given set, and so you don’t want to be restricted to considering only a particular,
concrete value for it.
To illustrate the first use, consider asking
Is there a number with the following property:
doubling it and adding 3 gives the same result as squaring it?
In this sentence you can introduce a variable to replace the potentially ambiguous
word “it”:
Is there a number 𝑥 with the property that 2𝑥 + 3 = 𝑥 2 ?
This material is intended solely for the academic utilization of St. Francis Xavier College.
2
CHAPTER 2: SPEAKING MATHEMATICALLY
The advantage of using a variable is that it allows you to give a temporary name
to what you are seeking so that you can perform concrete computations with it to
help discover its possible values.
To illustrate the second use of variables, consider the statement:
No matter what number might be chosen, if it is greater than 2,
then its square is greater than 4.
In this case introducing a variable to give a temporary name to the (arbitrary)
number you might choose enables you to maintain the generality of the statement,
and replacing all the instances of the word “it” by the name of the variable ensures
that possible ambiguity is avoided:
No matter what number 𝑛 might be chosen, if 𝑛 is greater than 2,
then 𝑛2 is greater than 4.
EXAMPLE
Writing Sentences Using Variables
Use variables to rewrite the following sentences more formally.
a. Are there numbers with the property that the sum of their squares equals the
square of their sum?
b. Given any real number, its square is nonnegative.
Solution:
a. Are there numbers 𝑎 and 𝑏 with the property that 𝑎2 + 𝑏2 = (𝑎 + 𝑏)2 ?
Or: Are there numbers 𝑎 and 𝑏 such that 𝑎2 + 𝑏2 = (𝑎 + 𝑏)2 ?
Or: Do there exist any numbers 𝑎 and 𝑏 such that 𝑎2 + 𝑏 2 = (𝑎 + 𝑏)2 ?
b. Given any real number 𝑟, 𝑟 2 is nonnegative.
Or: For any real number 𝑟, 𝑟 2 ≥ 0.
Or: For all real numbers 𝑟, 𝑟 2 ≥ 0.
I
In part (a) the answer is yes. For instance, 𝑎 = 1 𝑎𝑛𝑑 𝑏 = 0
would work. Can you think of other numbers that would also work?
____________________________________________________________
____________________________________________________________
____________________________________________________________
This material is intended solely for the academic utilization of St. Francis Xavier College.
3
CHAPTER 2: SPEAKING MATHEMATICALLY
Three of the most important kinds of sentences in mathematics are
universal statements, conditional statements, and existential
statements:
 A universal statement says that a certain property is true for all
elements in a set.
(Example: All positive numbers are greater than zero)
 A conditional statement says that if one thing is true then some
other thing also has to be true.
(Example: If 378 is divisible by 18, then 378 is divisible by 6.)
 Given the property that may not be true, an existential statement
says that there is at least one thing for which the property is
true.
(Example: There is a prime number that is even.)
In later sections we will define each kind of statement carefully and discuss
all of them in detail. The aim here is for you to realize that combinations of these
statement can be expressed in a variety of different ways. One way uses ordinary,
everyday language and another expresses the statement using one or more
variables. The exercises are designed to help you start becoming comfortable in
translating from one way to another.
UNIVERSAL CONDITIONAL STATEMENTS

Universal statements contain some variation of the words “for all” and
conditional statements contain versions “if – then”. A universal conditional
statement is a statement that is both universal and conditional. Here is an
example: For all animals 𝑎, if 𝑎 is a dog, then 𝑎 is a mammal.

One of the most important facts about universal conditional statements is
that they can be written in ways that make them appear to be purely
universal or purely conditional. For example, the previous statement can be
written in a way that makes its conditional nature explicit but its universal
nature implicit:
If 𝑎 is a dog, then 𝑎 is a mammal
𝑂𝑟: If an animal is a dog, then the animal is a mammal.
This material is intended solely for the academic utilization of St. Francis Xavier College.
4
CHAPTER 2: SPEAKING MATHEMATICALLY

The statement can also be expressed so as to make its universal nature
explicit and its conditional nature implicit:
For all dogs 𝑎, 𝑎 is a mammal
𝑂𝑟: All dogs are mammals.
EXAMPLE
Rewriting a Universal Conditional Statement
Fill in the blanks to rewrite the following statement:
For all real numbers 𝒙, if 𝒙 is nonzero then 𝒙𝟐 is positive.
a.
b.
c.
d.
e.
If a real number is nonzero, then its square _____.
For all nonzero real numbers 𝑥, ____.
If 𝑥 ____, then ____.
The square of any nonzero real number is ____.
All nonzero real numbers have ____.
Solution:
a.
b.
c.
d.
e.
is positive
𝑥 2 is positive
is a nonzero real number; 𝑥 2 is positive
positive
positive squares (or: squares that are positive)
CHECK YOUR PROGRESS
Fill in the blanks to rewrite the following statement:
For all real numbers 𝒙, if 𝒙 is greater than 2, then 𝒙𝟐 is greater than 4.
a.
b.
c.
d.
e.
If a real number is greater than 2, then its square is _______________________.
For all real numbers greater than 2, ___________________________________.
If 𝑥 ___________________________, then _______________________________.
The square of any real number greater than 2 is ________________________.
All real numbers greater than 2 have __________________________________.
This material is intended solely for the academic utilization of St. Francis Xavier College.
5
CHAPTER 2: SPEAKING MATHEMATICALLY
UNIVERSAL EXISTENTIAL STATEMENTS

A universal existential statements is a statement that is universal because its
first part says that a certain property is true for all objects of a given type, and
it is existential because its second part asserts the existence of something.
For example: Every real number has an additive inverse.

In this statement the property “has an additive inverse” applies universally
to all real numbers. “Has an additive inverse” asserts the existence of
something --- an additive inverse --- for each real number. However, the
nature of additive inverse depends on the real number; different real
numbers have different additive inverses. Knowing that the additive inverse
is a real number, you can rewrite this statement in several ways, some less
formal and some more formal:
All real numbers have different additive inverses.
Or: For all real numbers 𝑟, there is an additive inverse for 𝑟.
Or: For all real numbers 𝑟, there is a real number 𝑠 such that 𝑠
is an additive inverse for 𝑟.
EXAMPLE
Rewriting a Universal Existential Statement
Fill in the blanks to rewrite the following statement: Every pot has a lid.
a. All pots ____.
b. For all pots 𝑃, there is ____.
c. For all pots 𝑃, there is a lid 𝐿 such that _____.
Solution:
a. have lids
b. a lid for 𝑃.
c. 𝐿 is a lid for 𝑃.
CHECK YOUR PROGRESS
Fill in the blanks to rewrite the following statement: All bottles have cap.
a. Every bottle _______________________________________________________.
b. For all bottles 𝐵, there _______________________________________________.
c. For all bottles 𝐵, there is a cap 𝐶 such that ______________________________.
This material is intended solely for the academic utilization of St. Francis Xavier College.
6
CHAPTER 2: SPEAKING MATHEMATICALLY
EXISTENTIAL UNIVERSAL STATEMENTS

An existential universal statement is a statement that is existential because
its part asserts that a certain object exists and is universal because its second
part says that the object satisfies a certain property for all things of a certain
kind. For example: There is positive integer that is less than or equal to every
positive integer.

This statement is true because the number one is positive integer, and it
satisfies the property of being less than or equal to every positive integer. We
can rewrite the statement in several ways, some less formal and some formal:
Some positive integer is less than or equal to every positive integer.
Or: There is positive integer 𝑚 that is less than or equal to every positive integer.
Or: There is positive integer 𝑚 such that every positive integer is greater
than or equal to 𝑚.
Or: There is a positive integer 𝑚 with the property that for all integers 𝑛, 𝑚 ≤ 𝑛.
EXAMPLE
Rewriting an Existential Universal Statement
Fill in the blanks to rewrite the following statement in three different ways:
There is a person in my class who is at least as old as every person in my class.
a. Some ____ is at least as old as ____.
b. There is a person 𝑝 in my class such that 𝑝 is _____.
c. There is a person 𝑝 in my class with the property that for every person 𝑞 in my
class, 𝑝 is _____.
Solution:
a. person in my class; every person in my class
b. at least as old as every person in my class
c. at least as old as 𝑞
CHECK YOUR PROGRESS
Fill in the blanks to rewrite the following statement in three different ways:
There is a bird in this flock that is a least as heavy as every bird in the flock.
a. Some _______________________ is at least as heavy as _________________________.
b. There is a bird 𝑏 in this flock such that 𝑏 is __________________________________.
c. There is a bird 𝑏 in this flock with the property that for every bird 𝑏 in the flock,
𝑏 is ____________________________________________.
This material is intended solely for the academic utilization of St. Francis Xavier College.
7
CHAPTER 2: SPEAKING MATHEMATICALLY
d.
e.
f.
Stretch your Mind!
In an A4-sized paper, answer the following.
In each of 1 – 6, fill in the blanks using variable or variables to rewrite the given
statement.
1. Is there a real number whose square is -1
a. Is there a real number ______ whose square is -1?
b. Does there exist _____ such that 𝑥 2 = −1?
2. Is there an integer that has a remainder of 2 when it is divided by 5 and
remainder of 3 when it is divided by 6?
a. Is there an integer 𝑛 such that 𝑛 has _____?
b. Does there exist ____ such that if 𝑛 is divided by 5 the remainder is 2
and if ____?
Note: there are integers with this property. Can you think one?
3. Given any two real numbers, there is a real number in between.
a. Given any two real numbers 𝑎 and 𝑏, there is a real number 𝑐 such
that 𝑐 is ____.
b. For any two ____, _____ such that 𝑎 < 𝑐 < 𝑏.
4. Given any real number, there is a real number that is greater.
a. Given any real number 𝑟, there is ____ s such that 𝑠 is ____.
b. For any ____, ____ such that 𝑠 > 𝑟.
5. The reciprocal of any positive real numbers is positive.
a. Given any real number 𝑟, if 𝑟,the reciprocal of ____.
b. For any real number 𝑟, if 𝑟 is _____, then _____.
c. If a real number 𝑟 ____, then ____.
6. The cube root of any negative real number is negative.
a. Given any real number 𝑠, the cube root of _____.
b. For any real number 𝑠, if 𝑠 is ____, then ____.
c. If a real number 𝑠 ____, then ____.
This material is intended solely for the academic utilization of St. Francis Xavier College.
8
CHAPTER 2: SPEAKING MATHEMATICALLY
In each of 7 – 12, fill in the blanks to rewrite the given statement.
7. For all objects 𝐽, if 𝐽 is a square then 𝐽 has four sides.
a. All squares _____.
b. Every square _____.
c. If an object is a square. Then it _____.
d. If 𝐽 ____, then 𝐽 ______.
e. For all squares 𝐽, _____.
8. For all equations 𝐸, if 𝐸 is quadratic then 𝐸 has at most two real solutions.
a. All quadratic equations ____.
b. Every quadratic equation _____.
c. If an equation is quadratic, then it ______.
d. If 𝐸 ______, then 𝐸 ____.
e. For all quadratics equations 𝐸, ______.
9. Every nonzero real number has a reciprocal.
a. All nonzero real numbers _____.
b. For all nonzero real numbers 𝑟, there is ____ for 𝑟.
c. For all nonzero real numbers 𝑟, there is a real number 𝑠 such that
____.
10. Every positive number has a positive square root.
a. All positive numbers _____.
b. For any positive number 𝑒, there is ____ for 𝑒.
c. For all positive numbers 𝑒, there is a positive number 𝑟 such that
____.
11. There is a real number whose product with every number leaves the number
unchanged.
a. Some ____ has the property that its ____.
b. There is a real number 𝑟 such that the product of 𝑟 ____.
c. There is a real number 𝑟 with the property that for every real
number 𝑠, ____.
12. There is a real number whose product with every real number equals zero.
a. Some ____ has the property that its ____.
b. There is a real number 𝑎 such that the product of 𝑎 ____.
c. There is a real number 𝑎 with the property for every real number 𝑏,
___-.
This material is intended solely for the academic utilization of St. Francis Xavier College.
9
CHAPTER 2: SPEAKING MATHEMATICALLY
LESSON
2.2
The Language of Sets
This lesson is designed for you to:
 explain the concept of sets.
 explore the language of sets.
Interactive Discussion
What is a SET?
A set is a well-defined collection of distinct objects, considered as an
object in its own right. The arrangement of the objects in the set does
not matter. A set may be denoted by placing its objects between a pair
of curly braces
Notation

If 𝑆 is a set, the notation 𝑥 ∈ 𝑆 means that 𝑥 is an element of 𝑆. The
notation 𝑥 ∉ 𝑆 means that 𝑥 is not an element of 𝑆. A set may be specified
using the set roster notation by writing all of its elements between braces.
For example,
{1, 2, 3} denotes the set whose element 1, 2, 3. A variation
of the notation is sometimes used to describe a very large set, as when we
write {𝟏, 𝟐, 𝟑, . . . , 𝟏𝟎𝟎} to refer to the set of all integers from 1 to 100. A
similar notation can also describe an infinite set, as when we
write {𝟏, 𝟐, 𝟑, . . . } to refer to the set of all positive integers.
(The symbol, … , is called an ellipsis and is read “and so forth.”)

The axiom of extension says that a set is completely determined by what its
elements are --- not the order in which they might be listed or the fact that
some elements might be listed more than once.
This material is intended solely for the academic utilization of St. Francis Xavier College.
10
CHAPTER 2: SPEAKING MATHEMATICALLY
EXAMPLE
Using the Set – Roster Notation
a. Let 𝐴 = {1, 2, 3}, 𝐵 = {3, 1, 2}, 𝑎𝑛𝑑 𝐶 = {1, 1, 2, 3, 3, 3}.
What are the elements of 𝐴, 𝐵, 𝑎𝑛𝑑 𝐶? How are 𝐴, 𝐵, 𝑎𝑛𝑑 𝐶 related?
b. Is {0} = 0?
c. How many elements are in the set {1, {1}}?
d. For each nonnegative integer 𝑛, let 𝑈𝑛 = {𝑛, −𝑛}. Find 𝑈1 , 𝑈2 , 𝑎𝑛𝑑 𝑈0
Solution:
a. A, B, and C have exactly the same three elements: 1, 2, 3.
Therefore A, B, and C are simply different ways to represent the same set.
b. {0} ≠ 0, because {0} is a set with one element, namely 0, whereas 0 is just the
symbol that represents the number zero.
c. The set {1, {1}} has two elements: 1 and the set whose only element is 1.
d. 𝑈1 = {1, −1}, 𝑈2 = {2, −2}, 𝑈0 = {0, −0} = {0, 0}
CHECK YOUR PROGRESS
a. Let 𝑋 = {𝑎, 𝑏, 𝑐 }, 𝑌 = {𝑎, 𝑐, 𝑏}, 𝑎𝑛𝑑 𝑍 = {𝑎, 𝑏, 𝑏, 𝑐, 𝑐}. What are the elements
of X, Y, Z? How are X, Y and Z related?
b. How many elements are in the set {𝑎, {𝑎, 𝑏}, {𝑎}}?
c. For each positive integer 𝑥, let 𝐴𝑥 = {𝑥, 𝑥 2 }. Find 𝐴1 , 𝐴2 , 𝐴3 .
Your Solution:
This material is intended solely for the academic utilization of St. Francis Xavier College.
11
CHAPTER 2: SPEAKING MATHEMATICALLY
Certain sets of numbers are so frequently referred to that they are given special
symbolic names. These are the summarized in the table below.
Symbol
Set
R
Set of all real numbers
Z
Set of all integers
Q
Set of all rational numbers, or quotients of integers

Addition of superscript + or – or the letters 𝑛𝑜𝑛𝑛𝑒𝑔 indicates that only the
positive or negative or nonnegative elements of the set, respectively, are to
be included.

Thus 𝑅+denotes the set of positive real numbers, and 𝑍 𝑛𝑜𝑛𝑛𝑒𝑔refers to the
set of nonnegative integers: 0, 1, 2, 3, 4, and so forth.

Some authors refer to the set of nonnegative integers as the set of whole
numbers and denote it as W, thus, W = { 0, 1, 2, 3, … }. Other authors call
only the positive integers natural numbers, thus, N = { 1, 2, 3, … }.

The set of real numbers is usually pictured as the set of all points on a line, as
shown below. The number 0 corresponds to a middle point, called the origin.

The set of real numbers is therefore divided into three parts: the set of
positive real numbers, the set of negative real numbers, and the number 0.
Note that 0 is neither positive nor negative.

The real number line is called continuous because it is imagined to have no
holes. The set of integers corresponds to a collection of points located at
fixed intervals along the real number line.
Labels are given for a few real numbers corresponding to points on the line
shown below:
The Real Number Line
This material is intended solely for the academic utilization of St. Francis Xavier College.
12
CHAPTER 2: SPEAKING MATHEMATICALLY
Another way to specify a set uses what is called set-builder notation.
Set-Builder Notation
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. We denote this set as follows:
{𝑥 ∈ 𝑆|𝑃(𝑥 )}
the set of all
such that
Occasionally we will write {𝑥|𝑃 (𝑥 )} without being specific about where the element
x comes from. It turns out that unrestricted use of this notation can lead to genuine
contradictions in set theory.
EXAMPLE
Using the Set-Builder Notation
Given that R denotes the set of all real numbers, Z as the set of all integers, and 𝑍 +
as the set of all positive integers, describe each of the following sets.
a. {𝑥 ∈ 𝑅| − 2 < 𝑥 < 5}
b. {𝑥 ∈ 𝑍| − 2 < 𝑥 < 5}
c. {𝑥 ∈ 𝑍 +| − 2 < 𝑥 < 5}
Solution:
a. {𝑥 ∈ 𝑅| − 2 < 𝑥 < 5} is the open interval of real numbers (strictly) between
-2 and 5. It is pictured as follows:
b. {𝑥 ∈ 𝑍| − 2 < 𝑥 < 5} is the set of all integers (strictly) between -2 and 5.
It is equal to the set {−1, 0, 1, 2, 3, 4}.
c. Since all the integers in 𝑍 + are positive, {𝑥 ∈ 𝑍 +|−2 < 𝑥 < 5} = {1, 2, 3, 4}
This material is intended solely for the academic utilization of St. Francis Xavier College.
13
CHAPTER 2: SPEAKING MATHEMATICALLY
CHECK YOUR PROGRESS
Given that R denotes the set of all real numbers, Z the set of all integers,
and 𝑍 the set of all negative integers, describe of the following sets.
−
a. {𝑥 ∈ 𝑅|−5 < 𝑥 < 1}
b. {𝑥 ∈ 𝑍|−5 < 𝑥 < 1}
c. {𝑥 ∈ 𝑍 −| − 5 < 𝑥 < 1}
Your Solution:
What is a Subset?
SUBSET
If 𝐴 and 𝐵 are sets, then 𝐴 is called a subset of 𝐵, written 𝐴 ⊆ 𝐵, if and only if,
every element of 𝐴 is also an element of 𝐵.
Symbolically:
𝐴 ⊆ 𝐵 means that for all elements 𝑥, if 𝑥 ∈ 𝐴 𝑡ℎ𝑒𝑛 𝑥 ∈ 𝐵.
The phrases 𝐴 is contained in B and B contains A are alternative ways of saying that
A is a subset of b.
This material is intended solely for the academic utilization of St. Francis Xavier College.
14
CHAPTER 2: SPEAKING MATHEMATICALLY
It follows from the definition of subset that for a set A not to be a subset of set B
means that there is at least one element of A that is not an element of B. Symbolically:
𝐴 ⊈ 𝐵 means that there is at least one element x such that 𝑥 ∈ 𝐴 𝑎𝑛𝑑 𝑥 ∉ 𝐵.
What is a Proper Subset?
Proper Subset
Let 𝐴 and 𝐵 be sets. A is a proper subset of B if, and only if, every element of A is
in B but there is at least one element of B that is not in A.
EXAMPLE
Subsets
Let 𝐴 = 𝑍 +, 𝐵 = {𝑛 ∈ 𝑍|0 ≤ 𝑛 ≤ 100}, 𝑎𝑛𝑑 𝐶 = {100, 200, 300, 400, 500}.
Evaluate whether each statement is true or false.
a.
b.
c.
d.
e.
𝐵⊆𝐴
C is a proper subset of A
C and B have at least one element in common
𝐶⊆𝐵
𝐶⊆𝐶
Solution:
a. False. Zero is not a positive integer. Thus zero is in B but zero is not in A,
and so 𝐵 ⊈ 𝐴.
b. True. Each element in C is a positive integer and, hence, is in A, but there are
elements in A that are not in C. For instance, 1 is in A and not in C.
c. True. For example, 100 is in both C and B.
d. False. For example, 100 is in both C and B.
e. True. Every elements in c is in C. In general, the definition of subset implies
that all sets are subsets of themselves.
This material is intended solely for the academic utilization of St. Francis Xavier College.
15
CHAPTER 2: SPEAKING MATHEMATICALLY
CHECK YOUR PROGRESS
2
Let 𝐴 = { 2, {2}, (√2) } , 𝐵 = { 2, {2}, {{2}} } and 𝐶 = { 2 }.
Evaluate whether each statement is true or false.
a.
b.
c.
d.
e.
𝐴⊆𝐵
𝐵⊆𝐴
A is a proper subset of B
𝐶⊆𝐵
C is a proper subset of A
EXAMPLE
Distinction between ∈ and ⊆
Which of the following are true statements?
a.
b.
c.
d.
e.
f.
2 ∈ {1, 2, 3}
{2} ∈ {1, 2, 3}
2 ⊆ {1, 2, 3}
{2} ⊆ {1, 2, 3}
{2} ⊆ {{1}, {2}}
{2} ∈ {{1}, {2}}
Solution:

Only (a), (d), and (f) are true.

For (b) to be true, the set {1, 2, 3} would have to contain the element {2}. But
the only elements of {1, 2, 3} are 1, 2, and 3, and 2 is not equal to {2}.
Hence (b) is false.

For (c) to be true, the number 2 would have to be a set and every element in
the set 2 would have to be an element of {1, 2, 3}. This is not the case, so (c)
is false.

For (e) to be true, every element in the set containing only the number 2
would have to be an element of the set whose elements are {1} and {2}.
But 2 is not equal to either {1} or {2}, and so (e) is false.
This material is intended solely for the academic utilization of St. Francis Xavier College.
16
CHAPTER 2: SPEAKING MATHEMATICALLY
CHECK YOUR PROGRESS
Which of the following are true statement?
a.
b.
c.
d.
e.
𝑥 ∈ {𝑥, 𝑦, 𝑧}
𝑥 ⊆ {{𝑥}, {𝑦}, {𝑧}}
𝑥 ⊆ {𝑥, 𝑦, 𝑧}
{𝑥 } ⊆ {{𝑥}, {𝑦}, {𝑧}}
{𝑥 } ∈ {𝑥, 𝑦, 𝑧}
What is an Ordered Pair?
Ordered Pair
Given elements 𝑎 and 𝑏, the symbol (𝑎, 𝑏) denotes the ordered pair consisting
of 𝑎 and 𝑏 together with the specification that 𝑎 is the first element of the pair
and 𝑏 is the second element. Two ordered pairs (𝑎, 𝑏) and (𝑐, 𝑑) are equal if, and
only if, 𝑎 = 𝑐 𝑎𝑛𝑑 𝑏 = 𝑑. Symbolically:
(𝑎, 𝑏) = (𝑐, 𝑑) means that 𝑎 = 𝑐 𝑎𝑛𝑑 𝑏 = 𝑑.
EXAMPLE
Ordered Pairs
a. Is (1, 2) = (2,1)?
5
1
b. Is (3, 10) = (√9, 2) ?
c. What is the first element of (1, 1)?
Solution:
a. No. by the definition of equality of ordered pairs,
(1,2) = (2,1) if, and only if, 1 = 2 and 2 = 1.
But 1 ≠ 2, and so the ordered pairs are not equal.
b. Yes. By definition of equality of ordered pairs,
5
1
(3, 10) = (√9, 2) if, and only if 3 = √9 𝑎𝑛𝑑
5
10
1
= 2.
Because these equations are both true, the ordered pairs are equal.
c. In the ordered pair (1, 1), the first and the second elements are both 1.
This material is intended solely for the academic utilization of St. Francis Xavier College.
17
CHAPTER 2: SPEAKING MATHEMATICALLY
CHECK YOUR PROGRESS
a. Is (0, 10) = (10, 0)?
b. Is (4, 33 ) = (22 , 27)?
c. What is the first element of (2, 5)?
What is a Cartesian Product?
Cartesian Product
Given sets A and B, the Cartesian product of A and B, denoted A x B and read “A
cross B,” is the set of all ordered pairs (a, b), where 𝑎 is in A and 𝑏 is in B.
Symbolically:
𝐴 × 𝐵 = {(𝑎, 𝑏)|𝑎 ∈ 𝐴 𝑎𝑛𝑑 𝑏 ∈ 𝐵}.
EXAMPLE
Cartesian Products
Let 𝐴 = {1, 2, 3} and 𝐵 = {𝑢, 𝑣}
a.
b.
c.
d.
Find A × B.
Find 𝐵 × 𝐴.
Find 𝐵 × 𝐵.
How many elements are in 𝐴 × 𝐵, 𝐵 × 𝐴, and 𝐵 × 𝐵?
Solution:
a. 𝐴 × 𝐵 = {(1, 𝑢), (2, 𝑢), (3, 𝑢), (1, 𝑣 ), (2, 𝑣 ), (3, 𝑣 )}
b. 𝐵 × 𝐴 = {(𝑢, 1), (𝑢, 2), (𝑢, 3), (𝑣, 1), (𝑣, 2), (𝑣, 3)}
c. 𝐵 × 𝐵 = {(𝑢, 𝑢), (𝑢, 𝑣 ), (𝑣, 𝑢), (𝑣, 𝑣 )}
d. 𝐴 × 𝐵 has six elements. Note that this is the number of elements in A times
the number of elements in B. B × A has six elements, the number of elements
in B times the number of elements in A. B × B has four elements, the number
of elements in B times the number of elements in B.
This material is intended solely for the academic utilization of St. Francis Xavier College.
18
CHAPTER 2: SPEAKING MATHEMATICALLY
CHECK YOUR PROGRESS
Let 𝑌 = {𝑎, 𝑏, 𝑐 } and 𝑍 = {1, 2}
a. Find 𝑌 × 𝑍
b. Find 𝑍 × 𝑌
c. Find 𝑌 × 𝑌
How many elements are in 𝑌 × 𝑍, 𝑍 × 𝑌 𝑎𝑛𝑑 𝑌 × 𝑌?
Your Solution:
The Cartesian Plane
The Cartesian Plane can be made by
multiplying the set of real numbers to
another set of real numbers. That is:
R × R is the set of all ordered pairs (x, y)
where both x and y are real numbers. If
horizontal and vertical axes are drawn on
a plane and a unit length is marked off,
then each ordered pair R × R corresponds
to a unique point in the plane, with the
first and second elements of the pair
indicating, respectively, the horizontal
and vertical positions of the point.
This material is intended solely for the academic utilization of St. Francis Xavier College.
19
CHAPTER 2: SPEAKING MATHEMATICALLY
a.
b.
c.
Stretch your Mind!
In an A4-sized paper, answer the following.
1. Which of the following sets are equal?
𝐴 = {𝑎, 𝑏, 𝑐, 𝑑 }
𝐵 = {𝑑, 𝑒, 𝑎, 𝑐 }
𝐶 = {𝑑, 𝑏, 𝑎, 𝑐 }
𝐷 = {𝑎, 𝑎, 𝑑, 𝑒, 𝑐, 𝑒}
2. a. Is 4 = {4}?
b. How many elements are in the set {3, 4, 3, 5}?
c. How many elements are in the set { 1, {1}, {1, {1}} }?
3. a. 2 ∈ {2}?
b. How many elements are in the set {2, 2, 2, 2}?
c. How many elements are in the set {0, {0}}?
d. Is {0} ∈ {{0}, {1}}?
e. Is 0 ∈ {{0}, {1}}?
4. For each integer 𝑛, let 𝑇𝑛 = {𝑛, 𝑛2 }. How many elements are in each of
𝑇2 , 𝑇−3, 𝑇1 , 𝑎𝑛𝑑 𝑇0 ? Justify your answers.
5. Let 𝐴 = {𝑐, 𝑑, 𝑓, 𝑔}, 𝐵 = {𝑓, 𝑗}, 𝑎𝑛𝑑 𝐶 = {𝑑, 𝑔}. Answer each of the following
questions. Give reason for your answers.
a. Is 𝐵 ⊆ 𝐴?
b. Is 𝐶 ⊆ 𝐴?
c. Is 𝐶 ⊆ 𝐶?
d. Is C a proper subset of A?
6. a. Is 3 ∈ {1, 2, 3}?
b. Is 1 ⊆ {1}?
c. Is {2} ∈ {1, 2}?
d. Is {3} ∈ {1, {2}, {3}}?
e. Is 1 ∈ {1}?
f. Is {2 ⊆ {1, {2}, {3}}
h. Is 1 ∈ {{1}, 2}?
This material is intended solely for the academic utilization of St. Francis Xavier College.
20
CHAPTER 2: SPEAKING MATHEMATICALLY
7. a. Is ((−2)2 , −22 ) = (−22 , (−2)2 )?
b. Is (5, −5) = (−5, 5)?
3
c. Is (8 − 9, √−1) = (−1, −1)?
−2
3
d. Is (−4 , (−2)3 ) = (6 , −8)?
8. Let 𝐴 = {𝑤, 𝑥, 𝑦, 𝑧} and 𝐵 = {𝑎, 𝑏}. Use the set – roster notation to write each
of the following sets and indicate the number of elements that are in each set:
a. 𝐴 × 𝐵
b. 𝐵 × 𝐴
c. 𝐴 × 𝐴
d. 𝐵 × 𝐵
9. Let 𝑆 = {2, 4, 6} and 𝑇 = {1, 3, 5}. Use the set – roster notation to write each of
the following sets and indicate the number of elements that are in each set:
a. 𝑆 × 𝑇
b. 𝑇 × 𝑆
c. 𝑆 × 𝑆
d. 𝑇 × 𝑇
This material is intended solely for the academic utilization of St. Francis Xavier College.
21
CHAPTER 2: SPEAKING MATHEMATICALLY
LESSON
Mathematical Symbols
2.3
This lesson is designed for you to:
 determine different mathematical symbols.
 construct sentences using math terms and symbols.
BASIC MATH SYMBOLS
Symbol
Symbol Name
Meaning / Definition
Example
=
equals sign
equality
5 = 2+35 is equal to 2+3
≠
not equal sign
inequality
5 ≠ 45 is not equal to 4
≈
approximately
equal
approximation
sin(0.01) ≈ 0.01,x ≈ y
means x is approximately
equal to y
>
strict inequality
greater than
5 > 45 is greater than 4
<
strict inequality
less than
4 < 54 is less than 5
≥
inequality
greater than or
equal to
5 ≥ 4,x ≥ y means x is
greater than or equal to y
≤
inequality
less than or equal to
4 ≤ 5,x ≤ y means x is less
than or equal to y
()
parenthesis
calculate expression
inside first
2 * (3+5) = 16
[]
brackets
calculate expression
inside first
[(1+2)*(1+5)] = 18
This material is intended solely for the academic utilization of St. Francis Xavier College.
22
CHAPTER 2: SPEAKING MATHEMATICALLY
+
plus sign
addition
1+1=2
−
minus sign
subtraction
2−1=1
±
plus - minus
both plus and minus
operations
3 ± 5 = 8 or -2
∓
minus - plus
both minus and plus
operations
3 ∓ 5 = -2 or 8
*
asterisk
multiplication
2*3=6
×
times sign
multiplication
2×3=6
multiplication
2⋅3=6
division
6÷2=3
⋅
÷
multiplication
dot
division sign /
obelus
/
division slash
division
6/2=3
mod
modulo
remainder calculation
7 mod 2 = 1
.
period
decimal point,
decimal separator
2.56 = 2+56/100
ab
power
exponent
23 = 8
a^b
caret
exponent
2^3=8
√a
square root
√a ⋅ √a = a
√9 = ±3
3
cube root
4
fourth root
n
√a
nth root (radical)
%
percent
1% = 1/100
10% × 30 = 3
‰
per-mille
1‰ = 1/1000 = 0.1%
10‰ × 30 = 0.3
ppm
per-million
1ppm = 1/1000000
10ppm × 30 = 0.0003
ppb
per-billion
1ppb = 1/1000000000
10ppb × 30 = 3×10-7
ppt
per-trillion
1ppt = 10-12
10ppt × 30 = 3×10-10
√a
√a
3
3
3
√a ⋅ √a ⋅ √a = a
4
4
4
4
√a⋅ √a⋅ √a⋅ √a= a
3
√8= 2
4
√16= ±2
for n=3, n√8 = 2
This material is intended solely for the academic utilization of St. Francis Xavier College.
23
CHAPTER 2: SPEAKING MATHEMATICALLY
ALGEBRA SYMBOLS
Symbol
Symbol Name
Meaning /
Definition
Example
x
x variable
unknown value
to find
when 2x = 4, then x = 2
≡
equivalence
identical to
x
x variable
unknown value
to find
when 2x = 4, then x = 2
~
approximately
equal
weak
approximation
11 ~ 10
≈
approximately
equal
approximation
sin(0.01) ≈ 0.01
∝
proportional to
proportional to
y ∝ x when y = kx, k constant
∞
lemniscate
infinity symbol
n/a
≪
much less than
much less than
1 ≪ 1000000
≫
much greater than
()
parentheses
[]
brackets
{}
braces
set
n/a
⌊x⌋
floor brackets
rounds number
to lower integer
⌊4.3⌋ = 4
⌈x⌉
ceiling brackets
rounds number
to upper integer
⌈4.3⌉ = 5
x!
exclamation mark
factorial
4! = 1*2*3*4 = 24
|x|
single vertical bar
absolute value
| -5 | = 5
much greater
than
calculate
expression inside
first
calculate
expression inside
first
1000000 ≫ 1
2 * (3+5) = 16
[(1+2)*(1+5)] = 18
This material is intended solely for the academic utilization of St. Francis Xavier College.
24
CHAPTER 2: SPEAKING MATHEMATICALLY
f (x)
function of x
maps values of x
to f(x)
f (x) = 3x+5
(f ∘ g)
function
composition
(f ∘ g) (x) =
f (g(x))
f (x)=3x,g(x)=x-1 ⇒
(f ∘ g)(x)=3(x-1)
(a,b)
open interval
(a,b) =
{x | a < x < b}
x∈ (2,6)
[a,b]
closed interval
[a,b] =
{x | a ≤ x ≤ b}
x ∈ [2,6]
∆
delta
change /
difference
∆t = t1 - t0
∆
discriminant
Δ = b2 - 4ac
∑
sigma
summation sum of all values
in range of series
∑∑
sigma
double
summation
∏
capital pi
product of all
values in range
of series
∏ xi=x1∙x2∙...∙xn
e
e constant /
Euler's number
e = 2.718281828...
e = lim (1+1/x)x ,
x→∞
γ
Euler-Mascheroni
constant
γ=
0.5772156649...
φ
golden ratio
𝜋
pi constant
golden ratio
constant
𝜋=
3.141592654...is
the ratio between
the
circumference
and diameter of
a circle
∑ xi= x1+x2+...+xn
c = 𝜋 ⋅d = 2⋅𝜋 ⋅r
This material is intended solely for the academic utilization of St. Francis Xavier College.
25
CHAPTER 2: SPEAKING MATHEMATICALLY
GEOMETRY SYMBOLS
Symbol
Symbol Name
Meaning /
Definition
Example
∠
angle
formed by two
rays
∠ABC = 30°
measured angle
ABC = 30°
spherical angle
AOB = 30°
∟
right angle
= 90°
α = 90°
°
degree
1 turn = 360°
α = 60°
deg
degree
1 turn = 360deg
α = 60deg
′
prime
arcminute,
1° = 60′
α = 60°59′
″
double prime
arcsecond,
1′ = 60″
α = 60°59′59″
line
infinite line
line segment
line from point
A to point B
ray
line that start
from point A
arc
arc from point A
to point B
= 60°
⊥
perpendicular
perpendicular
lines (90° angle)
AC ⊥ BC
∥
parallel
parallel lines
AB ∥ CD
≅
congruent to
~
similarity
Δ
triangle
AB
equivalence of
geometric
shapes and size
same shapes,
not same size
triangle shape
∆ABC ≅ ∆XYZ
∆ABC ~ ∆XYZ
ΔABC ≅ ΔBCD
This material is intended solely for the academic utilization of St. Francis Xavier College.
26
CHAPTER 2: SPEAKING MATHEMATICALLY
distance
between points
x and y
radians angle
unit
radians angle
unit
|x-y|
distance
| x-y | = 5
rad
radians
c
radians
grad
gradians / gons
grads angle unit
360° = 400 grad
g
gradians / gons
grads angle unit
360° = 400 g
360° = 2π rad
360° = 2π c
SET THEORY SYMBOLS
Symbol
Symbol Name
Meaning /
Definition
Example
{}
set
a collection of
elements
A = {3,7,9,14}, B = {9,14,28}
|
such that
so that
A⋂B
intersection
A⋃B
union
A⊆B
subset
A⊂B
proper subset /
strict subset
A⊄B
not subset
objects that
belong to set A
and set B
objects that
belong to set A
or set B
A is a subset of
B. set A is
included in set
B.
A is a subset of
B, but A is not
equal to B.
set A is not a
subset of set B
A = {x | x∈
, x<0}
A ⋂ B = {9,14}
A ⋃ B = {3,7,9,14,28}
{9,14,28} ⊆ {9,14,28}
{9,14} ⊂ {9,14,28}
Source: https://www.prodigygame.com/main-en/blog/exhaustive-list-of-mathsymbols-their-meaning-downloadable-chart-for-classroom/
This material is intended solely for the academic utilization of St. Francis Xavier College.
27
CHAPTER 2: SPEAKING MATHEMATICALLY
Write it Up!
Use different mathematical terms/ symbols to construct any kind of
letter (friendly letter, love letter, apology letter).
DEAR ___________________ ,
_____________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
This material is intended solely for the academic utilization of St. Francis Xavier College.
28
CHAPTER 2: SPEAKING MATHEMATICALLY
CHAPTER TEST
Directions: Write your answers in a separate sheet of paper, A4-sized and
attach to this page. Show your solutions.
A. Fill in the blanks using a variable or variables to rewrite the given statement.
1. Is there a real number whose square root is -1?
a. Is there a real number x such that _____?
b. Does there exist ____ such that √𝑥 = −1?
2. Given any real number, there is a real number that is lesser.
a. Given any real number r, there is _____ s such that s is ____.
b. For any ___, ____ such that 𝑠 < 𝑟.
B. Fill in the blanks to rewrite the given statement.
3. For all real number x, if x is an integer then x is a rational number.
a. If a real number is an integer, then _____.
b. For all integers x, _____
c. Is x _____, then _____.
d. All integers x are _____.
4. All real numbers have squares that are not equal to -1.
a. Every real number has _____.
b. For all real numbers r, there is ____ for r.
c. For all real numbers r, there is a real numbers s such that ____.
5. There is a positive integer whose square is equal to itself.
a. Some ___ has the property that its _____.
b. There is a real number r such that the square of r is _____.
c. There is a real number r with the property that for every real number
s _____.
6. a. Let A be the set containing all prime number less than 30. List down all the
element of A.
b. Is {2, 2} = {2, {2}}?
c. How many elements are in the set {𝑎, 𝑎, 𝑎, 𝑎, 𝑎}?
This material is intended solely for the academic utilization of St. Francis Xavier College.
29
CHAPTER 2: SPEAKING MATHEMATICALLY
7. Given that Z denotes the set of all integers and N the set of all natural
numbers, describe each of the following sets.
a. {𝑥 ∈ 𝑁|𝑥 ≤ 10 𝑎𝑛𝑑 𝑥 𝑖𝑠 𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒 𝑏𝑦 3}
b. {𝑥 ∈ 𝑍|𝑥 𝑖𝑠 𝑝𝑟𝑖𝑚𝑒 𝑎𝑛𝑑 𝑥 𝑖𝑠 𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒 𝑏𝑦 2}
c. {𝑥 ⊆ 𝑍|𝑥 2 = 4}
8. Let 𝐵 = {2, 4, 6, 8, 10}, 𝐶 = {4, 8, 10}, 𝑎𝑛𝑑 𝐷 = {𝑥|𝑥 𝑖𝑠 𝑒𝑣𝑒𝑛}.
following questions. Give reasons for your answers.
a. Is 𝐷 ⊆ 𝐵?
b. Is 𝐶 ⊆ 𝐷?
c. Is 𝐶 ⊆ 𝐵?
d. Is B a proper subset of D?
Answer
the
9. a. Is ((−1)2 , 12 ) = (12 , (−1)2 )? Explain.
1
3
4
12
b. Is (√16, ) = (4,
)? Explain.
c. Is (−22 , 0) = (−√16, 0)? Explain
10. Let 𝐴 = {1, 2, 3, 4} and 𝐵 = {0, 1}. Use the set roster notation to write each of
the following sets, and indicate the number of elements that are in each set:
a. A × B
b. B × A
c. A × A
d. B × B
For additional 5 points:
Write your MATH HUGOT.
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
Book Reference:
Aufman, R., Lockwood, J., Nation, R., et.al. (2018). Mathematics in the Modern
World. Rex Bookstore, Inc.
This material is intended solely for the academic utilization of St. Francis Xavier College.
30