Department of Mathematics and Physics MATH 1100

MODULE 21

Mathematical language and symbols

Overview
Mathematics has its own language which is used by mathematicians
to communicate ideas, theories and concepts. Understanding its language
and symbols is vital in learning mathematics. Knowing the language will help
students understand and solve mathematical problems easier.

In this module, students will be introduced to the basic mathematical

language needed to express a range of mathematical concepts. Basic
mathematical concepts such as sets, functions and relations, binary
operations, and elementary logic will also be discussed.

The main goal is to achieve quality education.

Time allotment: 2 weeks

Objectives:
Upon completion of this module, you are expected to:
1. Discuss the language, symbol and conventions of mathematics.
2. Explain the nature of Mathematics as a language.
3. Perform operations on Mathematical expressions correctly.
4. Acknowledge that mathematics is a useful language.

*These objectives are lifted from CHED’s CMO.48 (2017)

PRE-ASSESSMENT

1. What do you think is the difference between expression and sentence in

mathematical setting?
2. What are the two ways in writing or describing a set?
3. What are the five operations on sets?
4. Are all functions a relation?
5. What is a binary operation and what are its properties?
6. What are the logical connectives used in combining simple statements or
propositions in logic?

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This module is based from the book “Mathematics in the Modern World” by the Department of Mathematics and
Physics, CS, CLSU.

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 Department of Mathematics and Physics MATH 1100

1. MATHEMATICS AS A LANGUAGE

Language is important in person’s daily activities. People use language to create ideas
and express them to other people. Similarly in the field of mathematics, mathematical
language is used to express mathematical ideas and concepts. All language has their own
vocabulary, and mathematics is not at exception.

1.1 Characteristics of mathematical language

A. Precise
Mathematical expressions or statements are precise, it has its own distinct meaning.

Preciseness of mathematical expression or statements is best learned through

understanding the language of mathematics.

Example 1. Reducing the long English sentence,

“The number of boys in a class, denoted by 𝑏, is less than the number of

girls in a class, denoted by 𝑔.”

symbolically into
𝑏 < 𝑔
greatly simplifies the sentence.

The symbols retain the important and exact information and the context need only to be
referred to again when stating a solution.

B. Concise
The language of mathematics is concise because it uses symbols instead of spelled-
out words for shortness of statements.

Example 2. The English sentence

“Three plus eight equals eleven.”
is expressed simply in the language of mathematics as
3 + 8 = 11.

By the use of symbols, mathematical expressions become brief, and ambiguities are
avoided.

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 Department of Mathematics and Physics MATH 1100

C. Powerful
Mathematics is powerful because students can only perform well in problem solving
if they understand the language of mathematics.

To express mathematical ideas, students need to master particular requirements and

conventions. In this way, complex ideas may be expressed in a greatly simplified manner.
In other words, learning the language of mathematics empowers the students to be
efficient problem solvers. Students also gain confidence in talking about their
mathematical learning and articulate for themselves what else they need to learn. Overall,
mathematical language skills include the abilities to read with comprehension, to express
mathematical thoughts clearly, to reason logically, and to recognize and employ common
patterns of mathematical thought.

1.2 Expressions vs. Sentences

Mathematical language is composed of expressions and sentences.

An expression is any correct arrangement of mathematics symbols, used to

represent a mathematical object of interest; it does not state a complete thought and so
it does not make sense to ask if it is true or false.

Example 3.
An example of expression can be as simple as 10 + 13.

We could change the (+) to make different mathematical expressions such as 10 – 13,
(10)(13), or 10 ÷ 13.

Addition, subtraction, multiplication, and division are called operations. There are many
more operations that can be used in a mathematical expression, which usually includes
numbers, sets, functions, ordered pairs, matrices, and others.

The following table lists the some key words used to express the four main operations.

Mathematical Operations

Addition Subtraction Multiplication Division

Add Subtract Multiply Divide

Increased by Decreased by Product Quotient

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 Department of Mathematics and Physics MATH 1100

Plus Minus Times Shared

Sum Difference Twice Split between

Total Reduced by Of Divided by

More Less than

Usually, verbal phrases are translated into variable expressions to simplify them into
an equivalent form that usually involves fewer symbols and operations, or into a form
that is best suited to a current application, or into a preferred form or style

Some examples are shown below:

Verbal Phrase Variable Expression
The difference of 𝑥 and 𝑦 𝑥−𝑦
The sum of a number and ten 𝑥 + 10
A number increased by four 𝑥+4
Three more than a number 𝑥+3
Three less than a number 𝑥−3
A number minus three 𝑥−3
Six subtracted from a number 𝑥−6
Five times a number 5𝑥
Twice a number 2𝑥
12
The quotient of 12 and a number
x
1
One half of a number x
2
The sum of five times a number and twelve 5𝑥 + 12
The product of six and twice a number (6)(2𝑥)
The square of a number 𝑥2
The square root of a number √𝑥
The cube of a number 𝑥3
The sum of the cube of a number and four x3  4
The square of the sum of a number and eleven (𝑥 + 11)2

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Whenever possible, select a single variable to represent an unknown quantity. Then,

express related quantities in terms of the selected variable.

Look at the following examples.

For each relationship, select a variable to represent one quantity and state what that
variable represents. Then, express the second quantity in terms of the variable selected.

1. Two consecutive odd integers.

Let 𝑥 = smaller odd integer
𝑥 + 2 = bigger odd integer
If you are wondering why?
First, list some consecutive odd integers: … , −5, −3, −1, 1, 3, 5, 7, 9, …
How do you get the odd number next to the previous odd number? We add 2.

2. The tens digit of a two-digit number exceeds the units digit by 5.

Let 𝑥 = units digit.
𝑥 + 5 = tens digit
Here we are considering two quantities: the units digit and the tens digit of the two-digit
number. How are these two quantities related?

3. The length of a rectangle is thrice its width.

Let 𝑥 = width of the rectangle
3𝑥 = length of the rectangle

We are considering two quantities: the length and the width of a rectangle. How are
these two quantities related?

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 Department of Mathematics and Physics MATH 1100

A mathematical sentence is a correct arrangement of mathematical symbols stating

a complete thought. The most common mathematical statements or sentences are called
equations and inequalities.

A mathematical sentence is one that makes a statement about the relationship of two
expressions. These two expressions are written in symbols such as numbers and
variables, or a combination of both. The relationship of the two expressions is usually
stated by using symbols or words such as
 equals (=),
 greater than (>),
 greater than or equal (≥),
 less than (<), or
 less than or equal(≤).

The table below shows a comparison between expressions and sentences.

Expression Sentence
It is just a mathematical phrase; It is a complete mathematical statement
a part of a sentence. with a complete thought.
A student only simplifies an expression. A student solves a sentence.
It has no relation symbol. Involves a relation symbol such as
equality or inequality.
Examples:
1.Ten is a number less than five:
1. A number less than five: 10 = 5 − 𝑥
5−𝑥 2. A number is less than five:
𝑥 < 5

The following table illustrates more on this.

Mathematical Expression in Words In Symbols

1. Thrice a number 𝑥 decreased by four is equal to ten more

3𝑥 − 4 = 𝑥 + 10
than the number 𝑥.

2. Half the difference of a number 𝑥 and its square is greater 𝑥 − 𝑥2

≥ −1
than or equal to negative one. 2

3. Twice a number 𝑥 less five is less than the sum of three and
2𝑥 − 5 < 3 + 𝑥
the number 𝑥.

4. Sum of the squares of a number 𝑥 and three is equal to the

𝑥 2 + 32 = 3 + 5𝑥
sum of three and the product of five and number 𝑥.

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In #4, “squares of a number 𝑥 and three” is written as 𝑥 2 + 32 . Observe that “squares”

is in plural form so it talks about the square of 𝑥 and the square of 3.

A mathematical sentence that is sometimes true or sometimes false is called an open

sentence. Open sentences usually arise when variables are used. To illustrate,

1. 𝑥 − 1 > 3𝑥 + 1
2. 2𝑥 − 18 = 6𝑦 + 1
3. A square has 𝑥 sides.

In the above illustrations, the mathematical sentences may or may not be true
depending on the values of the variables 𝑥 and 𝑦. The truth or falsity of such a sentence
is open, depending on the values of the variables.

On the other hand, we have a closed sentence if the mathematical sentence is

definitely true or definitely false. Each of the following are closed sentences. Why?

1. The smallest prime number is 1.

2. The square root of 9 is 3.
3. 𝑥 − 2 = 𝑥 + 2
4. −112 is an even number.
5. 663 + 336 = 999
6. The square root of – 16 is – 4.

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 Department of Mathematics and Physics MATH 1100

2. BASIC MATHEMATICAL CONCEPTS

Discussed below are fundamental concepts in mathematics; namely Sets, Functions and
Relations, Binary Operations, and Logic.

2.1 Sets
A set is a collection or grouping of elements. These elements can be anything such
as numbers, letters, names, sentences etc.

The capital letters 𝐴, 𝐵, 𝐶, … are usually used to name sets; if the elements are also letters,
the small letters 𝑎, 𝑏, 𝑐, … are used.

Describing sets

 Roster (or List) Method

 The simplest way of describing a set is to just list its elements separated by
commas inside a pair of braces.

It is easy to use especially if the set has only a few elements no matter what they are.

Example: The set 𝐴 whose elements are 𝑎, 𝑏, and 𝑐 can be expressed as:
𝐴 = {𝑎, 𝑏, 𝑐}

The order by which the elements are listed is irrelevant;

a set is defined by what elements it contains, not by any ordering or priority among those
elements. Thus, each of the following refers to the same set.
𝐴 = {𝑎, 𝑏, 𝑐 } 𝐴 = {𝑏, 𝑎, 𝑐 } 𝐴 = {𝑐, 𝑎, 𝑏}
𝐴 = {𝑎, 𝑐, 𝑏} 𝐴 = {𝑏, 𝑐, 𝑎} 𝐴 = {𝑐, 𝑏, 𝑎}

Other examples:
𝐵 = {𝑚, 𝑎, 𝑡, ℎ}
𝐶 = {CAg, CASS, CBAA, CEd, Cen, CHSI, CF, CS, CVSM}
𝐷 = {1, 2, 3, 4, 5}

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 Rule (or Description) Method

 Another way of describing a set is giving a description that befits each of the
elements.

Example:

𝐴 = {𝑥 | 𝑥 is a vowel of the English alphabet},

which is read as “𝐴 is composed of any 𝑥 , where 𝑥 is a vowel of the English alphabet”.

We call the number of elements of any set 𝐴 as the cardinal number of 𝐴. It is

denoted as |𝐴|.

Example: If 𝐴 = {𝑎, 𝑏, 𝑐, 𝑑, 𝑒, 𝑓, 𝑔, ℎ}, then |𝐴| = 8.

If an element 𝑥 is a member of the set 𝐴, we write 𝑥 ∈ A; otherwise, we write 𝑥 ∉ 𝐴.

Example: If 𝐴 = {𝑎, 𝑏, 𝑐, 𝑑, 𝑒, 𝑓, 𝑔, ℎ}, then 𝑓 ∈ 𝐴 but note that 𝐹 ∉ 𝐴.

Definition:
Set 𝐴 is a subset of set 𝐵, written 𝐴 ⊂ 𝐵, if every member of 𝐴 is also a member of 𝐵.
Otherwise, we write 𝐴 ⊄ 𝐵, read “𝐴 is not a subset of 𝐵” to mean there is at least one
element of 𝐴 that is not in 𝐵.

Example:
Given 𝐴 = {𝑎, 𝑏, 𝑐, 𝑑} and 𝐵 = {𝑎, 𝑏, 𝑐, 𝑑, 𝑒}.
Then 𝐴 ⊂ 𝐵 because all members of 𝐴 are members of 𝐵.
On the other hand, 𝐵 ⊄ 𝐴 because 𝑒 ∈ 𝐵 but 𝑒 ∉ 𝐵.

It follows from the definition that any set 𝐴 is a subset of itself, i.e., 𝐴 ⊂ 𝐴.

Note that ∈ and ⊂ mean two different concepts.

To illustrate,
𝑎 ∈ {𝑎, 𝑏} but 𝑎 ⊄ {𝑎, 𝑏}
{𝑎} ⊂ {𝑎, 𝑏} but {𝑎} ∉ {𝑎, 𝑏}
{𝑎} ∈ {{𝑎}, 𝑏, 𝑐} but {𝑎} ⊄ {{𝑎}, 𝑏, 𝑐}.
Definition:
Two sets are equal, written 𝐴 = 𝐵, if and only if they have the same elements.
Alternatively, 𝐴 = 𝐵, if and only if 𝐴 ⊂ 𝐵 and 𝐵 ⊂ 𝐴.

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Definition:
Any set that has no element at all is called a null (or empty) set, denoted by { } or 𝜙.

The null set is a subset of any other set.

The set 𝐴 = { 𝑥 | 𝑥 is an integer between 1 and 2} is a null set.

Definition:
Any set that contains all elements under consideration is called a universal set, denoted
by U.

Whenever necessary in any discussion, the universal set is always given or identified.

Operations On Sets

Given a list of sets, other sets may be formed by performing one or more operations on
the given sets. Basically these operations are the union (∪), intersection(∩), complement
(′), difference (−), and the Cartesian or cross product (×).

 Union
The union (∪) operation combines all elements of two sets.
Any element that occurs in both sets only occurs once in the new set.

Example:
If 𝐴 = {𝑎, 𝑏, 𝑐} and 𝐵 = {𝑐, 𝑑, 𝑒} then 𝐴 ∪ 𝐵 = {𝑎, 𝑏, 𝑐, 𝑑, 𝑒}.
If 𝐶 = {1, 2, 3, 5} and 𝐷 = {2, 4,6} then 𝐶 ∪ 𝐷 = ______________. (Answer2)

 Intersection
The intersection (∩) operation contains all elements found in two sets.
In other words, the intersection of two sets contains only the elements common to
both sets.

Example:
1. If 𝐴 = {𝑎, 𝑏, 𝑐} and 𝐵 = {𝑏, 𝑐, 𝑑, 𝑒} then
a. 𝐴 ∩ 𝐵 = {𝑏, 𝑐}
b. (𝐴 ∩ 𝐵) ∪ 𝐵 = {𝑐, 𝑑, 𝑒}

2. If 𝐶 = {1, 2, 3, 5} and 𝐷 = {2, 4,6} then 𝐶 ∩ 𝐷 = ______________. (Answer3)

3. If 𝐴 = {𝑎, 𝑏, 𝑐} and 𝐶 = {1, 2, 3, 5} then 𝐴 ∩ 𝐶 = { }.

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𝐶 ∪ 𝐷 = {1, 2, 3,4,5,6}
3
𝐶 ∩ 𝐷 = {2}

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 Department of Mathematics and Physics MATH 1100

 Complement
The complement (′) of a set, denoted 𝐴’, identifies the elements of the universal set
𝑈 that are not in 𝐴.

Examples:
1. If 𝐴 = {𝑥, 𝑦} and 𝑈 = {𝑥, 𝑦, 𝑧}, then 𝐴’ = {𝑧}.
2. If 𝐶 = {2, 4, 6, 8} and 𝑈 = {1, 2, 3, 4, 5, 6, 7, 8, 9} then 𝐶 ′ = _______. (Answer4)
3. Ø’ = 𝑈

 Difference
The difference of two sets 𝐴 and 𝐵, denoted 𝐴 − 𝐵, is defined to be the set whose
elements are those of 𝐴 that are not in 𝐵.

Example:
If 𝐴 = {𝑎, 𝑏, 𝑐} and 𝐵 = {𝑐, 𝑑, 𝑒}, then
a. 𝐴 − 𝐵 = {𝑎, 𝑏}
b. 𝐵 – 𝐴 = {𝑑, 𝑒}.

For the complement and difference operations,

𝐴′ = 𝑈 − 𝐴
𝐴 − 𝐵 = 𝐴 ∩ 𝐵′

 Cartesian Product
The Cartesian Product or Cross Product of two sets 𝐴 and 𝐵, denoted 𝐴 × 𝐵, is
the set of all ordered pairs (𝑥, 𝑦), such that 𝑥 ∈ 𝐴 and 𝑦 ∈ 𝐵.

Example:
If 𝐴 = {𝑎, 𝑏} and 𝐵 = {1, 2, 3}, then
a. 𝐴 × 𝐵 = {(𝑎, 1), (𝑎, 2), (𝑎, 3), (𝑏, 1), (𝑏, 2), (𝑏, 3)}
b. 𝐵 × 𝐴 = {(1, 𝑎), (1, 𝑏), (2, 𝑎), (2, 𝑏), (3, 𝑎), (3, 𝑏)}
Note: (3, 𝑎) ∉ 𝐴 × 𝐵

SUPPLEMENTARY VIDEOS:
For better understanding of the topics on sets, video links are provided below.
Iba't ibang Pamamaraan sa Pagsulat ng Set Notation
Algebra - Basic Set Notation - YouTube
Paano Makuha ang Subset ng isang Given Set - YouTube

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𝐶′ = {1, 3, 5, 7, 9}

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 Department of Mathematics and Physics MATH 1100

2.2 Functions and Relations

A relation 𝑅 is any set of ordered pairs (𝑥, 𝑦).

The set of all 𝑥-components is called the domain while the set of all 𝑦-components is
called the range.

Example:
The set 𝑅 = {(1, 2), (3, 4), (5, 6)} is a relation.
The first components 1, 3, 5 of the ordered pairs are respectively related to the second
components 2, 4, 6. The set {1, 3, 5} is called the domain of the relation while the set
{2, 4, 6} is called its range.

Consider the relation 𝐴 = {(1, 𝑎), (2, 𝑏), (3, 𝑏), (2, 𝑑), (1, 𝑒)}.
What is its domain? How about its range? (Answer5)

A function is a special kind of relation. In a function, every element of the domain is

related with one and only one element of the range . This means that no two distinct
ordered pairs have the same first components.

Illustrations:
1. The relation 𝐵 = {(1,2), (2,3), (3,4), (4,5), (5,6)} is a function.
2. The relation 𝐶 = {(1,2), (𝟐, 𝟑), (4,5), (𝟐, 𝟕)} is not a function.
Ordered pairs (2,3) and (2,7) have the same first components.

Aside from observing functions and relations in sets of ordered pairs, they may also
be described in other ways such as in graphs, and most often in equations that specify
the relationship between two variables 𝑥 and 𝑦.

In equations, functions are exclusively identified with the notation

𝑦 = 𝑓(𝑥).
With this notation, the function is given the name 𝑓; 𝑥 is called the independent
variable (also called argument) and 𝑦 is the dependent variable. The value of 𝑦 depends
upon the value of 𝑥.

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Domain: {1, 2, 3} ; Range: {𝑎, 𝑏, 𝑑, 𝑒}

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The function 𝑓 may be likened into a machine where if

we input 𝑥 into it, it produces the output 𝑦.

Say, the machine (function 𝑓) is a juice-maker. If we put in a mango, then mango

juice will come out; if we put in a pineapple, then pineapple juice comes out. But if we
put in a stone (not in its domain!), what happens? Functions are similar to the machine.

Now suppose we have two functions 𝑓 and 𝑔 defined by

𝑓(𝑥) = 2𝑥 + 3 and 𝑔(𝑥) = 𝑥 2 – 3𝑥 + 2.

Then, 𝑓(7) means inputting 7 into function f which processes it accordingly as 2(7) + 3
that produces the output 17.

Similarly, 𝑔(−4) means inputting −4 into function 𝑔 which processes it accordingly as

(– 4)2 – 3(– 4) + 2 and produces the output 30.

Depending on how the function is defined (how the machine is designed), inputs
(expressions) are processed accordingly to produce an output.

Example: Consider 𝑓(𝑥) = 3𝑥 2 – 5𝑥 + 2 and 𝑔(𝑥) = 2 – 3𝑥. Find each of the following:
a. 𝑓(2) d. 𝑓(1/2) – 𝑔(2) + 3/2
b. 𝑔(– 3/2) e. 𝑓(3) + 𝑔(2𝑥– 7)
c. 𝑓(– 2) – 𝑔(4/3) f. 𝑓(𝑔(𝑥))
Solutions.
a. 𝑓(2) = 3(2)2 – 5(2) + 2 = 3(4) – 10 + 2 = 4

3 3 9 13
b. 𝑔 (– ) = 2 – 3 (– ) = 2 + =
2 2 2 2

4 4
c. 𝑓 (– 2) − 𝑔 ( ) = [3(– 2)2 – 5(– 2) + 2] − [2 – 3 ( )]
3 3
= [12 + 10 + 2] – [2 – 4]
= 24 – (– 2)
= 26

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1 2 1 2 1 3
d. 𝑓 ( ) – 𝑔(2) + = [3 ( ) – 5 ( ) + 2] – [2 – 3(2)] +
2 3 2 2 2
3 5 3
= [ – + 2] – [2 – 6] +
4 2 2
1 3
= – [– 4] +
4 2
23
=
4

𝑓(3) + 𝑔(2𝑥– 7) = [3(3)2 – 5(3) + 2] + [2 – 3(2𝑥– 7)]

e.
= [14] + [2 – 6𝑥 + 21]
= 37 − 6𝑥

f. 𝑓(𝑔(𝑥 )) = 3[𝑔(𝑥 )]2 – 5[𝑔(𝑥 )] + 2

= 3(2 – 3𝑥 )2 – 5(2 – 3𝑥) + 2
= 3(4 – 12𝑥 + 9𝑥2) – 10 + 15𝑥 + 2
= 27𝑥 2 – 21𝑥 + 4

SUPPLEMENTARY VIDEOS:
For better understanding on operations and compositions of functions, video links are
provided below.

Algebra - Function Operations in Filipino - YouTube

Composition of Functions in Filipino | ALGEBRA | PAANO | COMPOSITE - YouTube

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 Department of Mathematics and Physics MATH 1100

2.3 Binary Operations

A binary operation on a set is a rule for combining two elements of the set, to produce
another element of the same set. A binary operation ＊ defined on a non-empty set 𝑆 is
a rule that assigns to each ordered pair (𝑎, 𝑏) of elements of 𝑆 a unique element 𝑎＊𝑏 ∈
𝑆.

Illustrations:
1. On the set of natural numbers 𝑁 = {1, 2, 3, 4, 5, . . . }, addition and multiplication
are binary operations because adding or multiplying any two elements of 𝑁
produces an element that also belongs to 𝑁.

2. On the set of natural numbers 𝑁 = {1, 2, 3, 4, 5, . . . }, subtraction and division are

not binary operations. Why? (Answer6)

Properties of a Binary Operation

1. Closure Property
The binary operation ＊ defined on a set 𝐴 is closed on set 𝐴 if and only if
𝒂＊𝒃 ∈ 𝑨 for any 𝒂, 𝒃 ∈ 𝑨.

Example:
The binary operation addition “+” is closed on the set of integers because when we add
any two integers, the result is also an integer.

However, division “/” is not closed on the set of integers because when we divide any
two integers such as 5/3, the result may happen to be not integer.

2. Commutative Property
A binary operation＊ defined on a set A is said to be commutative if
𝒂＊𝒃 = 𝒃＊𝒂 for any 𝒂, 𝒃 ∈ 𝑨.

Example:
The binary operations addition and multiplication are commutative on the set of real
numbers.

Subtraction and division are not commutative because for example,

2– 5 ≠ 5– 2 and 8/4 ≠ 4/8.

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If you consider two elements in 𝑁, say 1 and 5, 1 − 5 = −4 or 1 ÷ 5 = 1/5 is not an element of 𝑆. So subtraction
and division are not binary operations in 𝑁.

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 Department of Mathematics and Physics MATH 1100

3. Associative Property
A binary operation ＊ defined on a set 𝐴 is said to be associative if
(𝒂＊𝒃)＊𝒄 = 𝒂＊(𝒃＊𝒄) for any 𝒂, 𝒃, 𝒄 ∈ 𝑨.

Example:
Both addition and multiplication are associative operations on natural numbers.
Subtraction is not an associative operation because for example,
(11 – 18) – 7 ≠ 11 – (18 – 7) .

4. Existence of an Identity Element

If ＊ is a binary operation on 𝐴, an element 𝑚 ∈ 𝐴 is an identity element of 𝐴 with
respect to ＊ if
𝒂 ＊ 𝒎 = 𝒎 ＊ 𝒂 = 𝒂 for any 𝒂 ∈ 𝑨

Example:
In the set of real numbers, the identity element for multiplication is 1 while the identity
element for addition is 0.

5. Existence of an Inverse Element

Let ＊ be a binary operation on 𝐴 with identity 𝑚, and let 𝑎 ∈ 𝐴.
If there exists an element 𝑏 ∈ 𝐴 such that 𝒂＊𝒃 = 𝒃＊𝒂 = 𝒎, then 𝑏 is called the
inverse element of 𝑎.

Note: An inverse may or may not exist for some elements.

Example:
In the set of integers ℤ, every element has an additive inverse.
However, not one of the elements has a multiplicative inverse.

In the set of whole numbers 𝑊 = {0, 1, 2, 3, 4, . . . }, the only element that has an additive
inverse is zero; its inverse is itself.

SAQ: Suppose ＊is defined on the set of integers ℤ by 𝑎＊ 𝑏 = √𝑎𝑏 − 3 . Find

a. 4＊ 3 b. – 3 ＊– 13 c. 0 ＊4 d. Is ＊ a binary operation? Why?
Solutions:
a. 4＊3 = √4(3) − 3 = √12 − 3 = √9 = 3
b. – 3＊– 13 = √(−3)(−13) − 3 = √39 − 3 = √36 = 6
c. 0＊4 = √0(4) − 3 = √−3 = ? , does not exists
d. Item (c) shows that ＊ is not a binary operation.

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2.4 Elementary Logic

Logic is the primary basis of all mathematical reasoning. As such, it is concerned with the
investigation of consequences that hold between the premises and the conclusion of a
sound argument.

An argument is said to be sound (valid, correct) if its conclusions follows from, or is a

consequence of its premises; otherwise it is unsound.

Statements (Propositions)
A statement (or proposition) is the basic building block of logical reasoning. It is a
declarative sentence that is either true or false, but not both. If it is true, its truth value
is said to be T (true); otherwise, it is an F (false).

Illustrations:
1. “10 + 12 = 22” is a statement that is true.
2. “The letter s is a vowel” is a statement that is false.
3. “Samsung creates cellphones” is a statement that is true.
4. “Samsung creates the best cellphones” is not a statement;
it is an opinion that may be true to some people but false to others.
5. “2𝑥 + 5 = 𝑦 − 3” is not a statement;
it is sometimes true and sometimes false.

Statements are commonly represented by small letters; most frequently used are p, q, r,
s, and t.

For example, the statement “It is raining” may be represented by p as in the following:
p: It is raining.

If it is truly raining, its truth value is T; if not, it is F.

Compound Statements

Statements constructed by connecting one or more simpler statements are

called compound statements. The simpler statements are connected together by
using connectives such as “not”, ‘‘and”, “or”, “if…then”, “only if”, and “if and only if”.

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Example:
“Ben donated ₱10,000 and he did not take a bath or attended his class”
is a compound statement that is composed of 3 simpler statements:
p: Ben donated P10000.
q: He did not take a bath.
r: He attended his class.

To determine the truth value of a compound statement, we need to first consider and
examine the way the simpler statements are connected. The resulting compound
statement can be a negation, a conjunction, a disjunction, an implication, a double
implication or any combination of these.

Negation (¬)
The negation of a statement is denial of the statement. To negate a statement, the
symbol ¬ is used. To illustrate,

Statement Negation
p: It is raining. ¬p: It is not raining.

q: The land is wet. ¬q: The land is not wet.

r: Some classmates are upperclassmen. ¬r: No classmates are upperclassmen.

Note:
“Some classmates are not upperclassmen”
is not a negation of r (why?)

s: All CLSU students are optimistic. ¬s: Not all CLSU students are optimistic.
Note:
“All CLSU students are not optimistic.”
is not a negation of s (why?)

The statement “r: Some classmates are upperclassmen.” means that at least one of the
classmate is upperclassman. The negation of the statement should mean that none of
the classmates are upperclassmen.

The statement “s: All CLSU students are optimistic.” means that every CLSU student is
optimistic. The negation of the statement should mean that there is at least one CLSU
student that is not optimistic.

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A negation is true T if and only the statement itself is false F. Similarly, a negation is false
F if and only the statement itself is true T.

Conjunction (∧)
A conjunction consists of 2 or more simpler statements that are connected by the word
“and ”, represented by the symbol ∧. That is, the conjunction p ∧ q represents the
compound statement “p and q”.

Illustration: If p: Janre is a CLSU graduate.

q: Janre is 30 years old.
Then, p ∧ q: Janre is a CLSU graduate and Janre is 31 years old.
p ∧ ¬q: Janre is a CLSU graduate and Janre is not 31 years old.
¬p ∧ q: Janre is not a CLSU graduate and Janre is 31 years old.
¬p ∧ ¬q: Janre is not a CLSU graduate and Janre is not 31 years old

A conjunction of 2 statements p and q is true if and only if both p and q are true.

Disjunction (∨)
A disjunction consists of 2 or more simpler statements that are connected by the word
“or ”, represented by the symbol ∨. That is, the conjunction p ∨ q represents the
compound statement “p or q”.

Illustration: If p: Tom is a BSMath student.

q: Jerry is a BSAgri student.
Then, p ∨ q: Tom is a BSMath student or Jerry is a BSAgri student.
¬p ∨ q: Tom is not a BSMath student or Jerry is a BSAgri student.

A disjunction of 2 statements p and q is true if and only if at least one of p and q is true.

 A popular compound statement, known as De Morgan’s Law, is the equivalent

of negating a conjunction or negating a disjunction. Namely,
i. ¬(p ∧ q) = ¬p ∨ ¬q
ii. ¬ (p ∨ q) = ¬p ∧ ¬q

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Implication (  )
An implication (also called conditional) consists of two cause-and-effect statements. One
is a premise and the other is a consequence that are connected by the words “if…then”.
The symbol  is used. That is, the implication p  q represents the compound
statement “if p then q”. Equivalently, it means

 "p is sufficient for q"

 "q when p"
 "a necessary condition for q is p"
 "q unless not p"
 "q follows from p

Illustration: p: It is raining.
q: The ground is wet.
p  q: If it is raining then the ground is wet.
¬p  q: If it is not raining then the ground is wet.

An implication p  q is considered false only if p is true and q is false.

Sometimes, we are interested in taking the converse, the inverse, or the contrapositive
of an implication p  q. These are defined as

Converse : q  p
Inverse: ¬p  ¬q
Contrapositive: ¬q  ¬p

Example:
Statement: If you are more than 60 years old, then you are entitled to a senior
citizen’s card.
Converse: If you are entitled to a senior citizen’s card, then you are more than 60
years old.
Inverse: If you are not more than 60 years old, then you are not entitled to a
senior citizen’s card.
Contrapositive: If you are not entitled to a senior citizen’s card, then you are not more
than 60 years old.

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Double Implication (  )
A double implication (also called biconditional) is a statement of the form
(p  q)  (q  p),
and is written as p  q.

The double implication p  q is read “p if and only if q” and is frequently abbreviated “p

iff q”. It is a conjunction of the two implications p  q and q  p which are conversely
related; one statement is a necessary and sufficient condition for the other.

Illustration: p: ∆ABC is a right triangle.

q: c2 = a2 + b2 .
p  q: ∆ABC is a right triangle iff c2 = a2 + b2 .
: c2 = a2 + b2 iff ∆ABC is a right triangle.

A biconditional p  q is considered true if and only if p and q are both true or are both
false.

SUMMARY

 The three characteristics of mathematical language are precise, concise and

powerful.
 An expression is just a mathematical phrase; a part of a sentence while sentence
is complete mathematical statement with a complete thought.
 There are two types of sentences; open sentences (mathematical sentence that
is sometimes true or sometimes false) and closed sentence (mathematical
sentence that is definitely true or definitely false).
 A set is collection or grouping of elements which can be written in roster and rule
method.
 The five operations on sets are union, intersection, complement, difference, and
Cartesian product.
 A function is a special kind of relation with 𝑥-components called domain and 𝑦-
components called range.
 A binary operation on a set is a rule for combining two elements of the set, to
produce another element of the same set with the following properties: closure,
commutative, associative, existence of an identity element, and existence of an
inverse element.
 A statement (or proposition) is a declarative sentence that is either true or false,
but not both.

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 Compound statements are simpler statements connected together by using logical

connectives such as “not”, ‘‘and”, “or”, “if…then”, “only if”, and “if and only if”.
 Compound statement can be a negation, a conjunction, a disjunction, an
implication, a double implication or any combination of these.

POST ASSESSMENT:

I. Tell whether the given is an expression or a sentence.

1. 3𝑥 – 4𝑦
2. (Ø, – 5)
3. 4𝑥 – 3 = 5
4. 3𝑥 < 6
2 0
5.  
 1 7

II. Translate the given phrase into a mathematical expression.

1. The sum of twice a number x and another number y.
2. Twice the sum of two numbers x and y.
3. The age of Ted 3 years ago if he is x years old now.
4. The product of 3 consecutive odd numbers if the middle number is x.
5. The length of a rectangle if it is 9 units longer than half its width w.

III. Translate the given mathematical expressions into a verbal phrase.

1. 3𝑥 – 1
2. (𝑥 – 5)(𝑥 + 5)
3. 1 + 2 + 3 + … + 10

IV. (Sets) Perform the indicated operations on sets.

Consider U = (0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {0, 2, 4, 6, 8},
B = {1, 3, 5, 7, 9} and C= {3, 4, 5, 6, 7}.
Find: 1. (A  B)  C

2. A  (C  B)

3. (A  C ')  (B  C )

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V. (Functions) Suppose f(x) = 2x – 9 and g(x) = 3x2 – 5x + 2. Find the following:

1. f(3/2) + g(–9)
f (2) 1
2. 
g (1) 2

3. f(2x – 3) – g(x + 4)

VI. Binary Operations

1. If x＊ y = 5y – 3x, find a. 3＊ 2 b. –4＊3/4 c. –4＊b/2
2. Which of the following definitions of ＊is commutative?
𝑥+𝑦
a. x＊y = 3x + 2y b. x＊y = xy c. x＊y = 𝑥𝑦 d. x＊y = y – 2x

3. Given x＊y = 2x + 3y, what is the value of 3＊(4＊5) ?

VII. Elementary Logic

1. Which of the following are statements? Not statements? Why?
a. When is your birthday?
b. I will pass in this subject.
c. The number 3.2 is an even number.

2. Suppose p: The sun is shining; q: It is raining; and r: The ground is wet.

Write each of the following in symbolic forms.
a. If it is raining, then the sun is not shining.
b. It is raining and the ground is wet.
c. The ground is not wet.
d. Either the sun is shining or it is raining.
e. The ground is wet if and only if it is raining and the sun is not shining.

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REFERENCES:

Badua, P. M., Daquioag, A. Z., Daquioag, R. R., Florendo, D. R., Ibanez, E. D., Pagay,
A. M., Romano, M. G., Seeping, A. V., Taganap, E. C., & Tumaliuan, M. N. (2012).
Functions, Relations and Graphs. In Pre-Calculus Math (Plane Trigonometry) (pp. 62-
75). Love Printing Press, Cabanatuan City.

Badua, P. M., Daquioag, A. Z., Daquioag, R. R., Florendo, D. R., Ibanez, E. D., Pagay,
A. M., Romano, M. G., Seeping, A. V., Taganap, E. C., & Tumaliuan, M. N. (2012).
Sets. In Pre-Calculus Math (College Algebra) (pp. 1-24). Love Printing Press,
Cabanatuan City.

Aufman, R. N., Lockwood, J., & Richard, D. (2013). Logic. In Mathematical Excursions
(3rd ed., pp. 111-119). Brooks/Cole, Cengage Learning.

Aufman, R. N., Lockwood, J., & Richard, D. (2013). Logic. In Mathematical Excursions
(3rd ed., pp. 133-137). Brooks/Cole, Cengage Learning.

Aufman, R. N., Lockwood, J., & Richard, D. (2013). Logic. In Mathematical Excursions
(3rd ed., pp. 141-144). Brooks/Cole, Cengage Learning.

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