I.

Characteristics of Mathematical Language

Mathematics, as a body of knowledge, has a structure and all elements and operations in it, so we can say
that mathematical language is a system used to communicate mathematical ideas. It consists of some
natural language using technical terms (mathematical terms) and grammatical conventions, supplemented
by a highly specialized symbolic notation for mathematical formulas. This maybe not apparent to you as
student because you are often overwhelmed by topics discussed to you in math subjects. You may be
forgetting the basic concepts in mathematics that is very important as a foundation in all mathematical
learnings.

Mathematical language is your tool to communicate mathematical ideas easier because of the following
characteristics.
1. Math language is non-temporal. It has no past, present and future tense unlike in English
language. There is no conjugation of words and mathematics statements are presented
simple as “is”. Math language carries no emotional content.
2. It has no equivalent words for joy or sadness. Your aesthetic experience like mathematicians
about math is only a subjective experience. Good values are learned thru mathematics but
cannot be found in a mathematical language.
3. Math language is precise. Statements are exact and accurate. As you can observe math
language is clearly stated and lacks uncertainty.
4. Math language is concise. No need for unnecessary words and briefly stated.
5. Math language is powerful. Complex ideas are well expressed

Parts of Speech in Mathematics

1
1. Numbers or Constants 2, 4, 2 , 5
2. Variables 𝑥, 𝑦, 𝑧
3. Relation Symbols =, ≤, ≥,∪,∩, ∈, ⊂
4. Operation Symbols +, −,×,÷
5. Grouping Symbols ( ), { }, [ ]

II. Expressions vs Sentences

Every language has its vocabulary (the words) and its rules for combining these words into complete
thoughts (the sentences). Mathematics is no exception. In studying the mathematical language, we will
make a very broad classification between the ‘nouns’ of mathematics (used to name mathematical objects
of interest) and the ‘sentences’ of mathematics (which state complete mathematical thoughts).
 Mathematical Mathematical
Expression Sentence

Incomplete Complete
thought thought

Example: Example:
3x + 5 3x + 5 = 9
4y 4y > 2

Mathematical expression

An expression is the mathematical analogue of an English noun; it is a correct arrangement of

mathematical symbols used to represent a mathematical object of interest.

An expression does not state a complete thought; it does not make sense to ask if an expression is true or
false. The most common expression types are numbers, sets, and functions. Numbers have lots of different
names: for example, the expressions 5, 2 + 3, 10 + 2, (6 − 2) + 1, and 1 + 1 + 1 + 1 all look different,
but are all just different names for the same number. This simple idea—that numbers have lots of different
names—is extremely important in mathematics!

Mathematical sentence

A mathematical sentence is the analogue of an English sentence; it is a correct arrangement of

mathematical symbols that states a complete thought. Sentences have verbs. In the mathematical
sentence ‘3+4=7, 3+4=7’, the verb is ‘=’. A sentence can be (always) true, (always) false, or sometimes
true/sometimes false.

Examples:
1+2=3 True

1+2=4 False

𝑥=2 Sometime true/sometimes false

It is true when 𝑥 is 2, and false otherwise.

𝑥+3=3+𝑥 Always true, no matter what value is chosen for 𝑥

III. Conventions in Mathematical Language

Mathematics has its own language, much of which we are already familiar with. For example, the digits
0, 1, 2, 3, 4, 5, 6, 7, 8, 9 are part of our everyday lives. There are many symbols in mathematics and most
are used as a precise form of shorthand. We need to be confident when using these symbols, and to gain
that confidence we need to understand their meaning. To understand their meaning there are two things
to help us:
 ✓ context - this is the context in which we are working, or the particular topics being
studied, and
✓ convention - where mathematicians and scientists have decided that particular symbols
will have particular meaning.

Mathematical Convention is a fact, name, notation, or usage which is generally agreed upon by
mathematicians. PEMDAS (Parenthesis, Exponent, Multiplication, Division, Addition and Subtraction) is an
example. All mathematical names and symbols are conventional.

Examples of conventions:
• The letters used for the sides of a triangle are usually 𝑎, 𝑏, and 𝑐, where 𝑐 is the hypotenuse. The
capital letters 𝐴, 𝐵, and 𝐶 are used for the angles.
• Functions are denoted by 𝑓(𝑥), 𝑔(𝑥), or ℎ(𝑥).
• In writing an algebraic expression, the numerical coefficient is written before the variable in a term.
• The Greek letter ‘pi’, written π, is used to represent the number 3.14159....
• We often use α (‘alpha’), β (‘beta’), and θ (‘theta’) to represent angles.
• The Greek capital letter ‘sigma’ or Σ is frequently used to represent the addition of several numbers.

There are many other conventions that most mathematicians have been practicing over the years. It is not
to say that other formats are incorrect, but since they are already conventions, then it will be easier to use
them to avoid confusion.

Common Mathematical Symbols

Symbols save time and space when writing. These are three groups of the most commonly used math
symbols that you need to learn in order to solve math questions. (Take time to research on other examples.)

• Basic Math Symbols +, −,×,÷, =, ≠, ≥

• Set Theory Symbols ∈,∪,∩, ⊂, ⊆, ∉
• Logic symbols ¬, →, ^, ↔
IV. Sets, Functions, Relations, and Binary Operations

Sets
Any group or collection of objects is called a set. The objects that belong in a set are the elements, or
members of the set. For example, the set consisting of the four seasons has spring, summer, fall, and winter
as its elements.

A set is a well-defined collection of objects if it is possible to determine whether any given item is an
element of the set. For instance, the set of letters of the English alphabet is well defined. The set of great
songs is not a well-defined set. It is not possible to determine whether any given song is an element of the
set or is not an element of the set because there is no standard method for making such a judgment. The
statement “4 is an element of the set of natural numbers” can be written using mathematical notation as
4 ∈ ℕ. The symbol ∈ is read “is an element of”. To state that “−3 is not an element of the set of natural
numbers,” we use ∉ as the symbol for “is not an element of ”, and write -3 ∉ ℕ.

Example:
Determine whether each statement is true or false.
1. 4 ∈ {2,3,4,7} True, 4 is in the given set
2. −5 = {2,3,4,7} False, -5 is not in the given set
1 1
3. 2 ∉ 𝐼 True, 2 is not an integer
4. The set of nice cars is a well-defined set. False, nice is not precise.

Basic Number Sets

Natural Numbers or Counting Numbers ℕ = {1, 2, 3, 4, 5, . . . }

Integers ℤ = {. . . , −4, −3, −2, −1, 0, 1, 2, 3, 4, . . . }

𝑚
Rational Numbers ℚ = numbers that can be written in the form where 𝑚, 𝑛 ∈ ℤ
𝑛
𝐶
Irrational Numbers ℚ = numbers that are not rationals

Real Numbers ℝ = the set of all rational or irrational numbers

There are two ways to describe a set. In Listing Method, all or partial members of the set are listed. In Set-
Builder Method, the set is described by listing the properties that describe the elements of the set.

Examples:

Set Listing Method Set – Builder Method

Let 𝑨 be the set of natural

𝐴 = {1,2,3,4,5,6,7,8,9} 𝐴 = {𝑥|𝑥 ∈ ℕ, 𝑥 < 10}
numbers less than 10.

Let 𝑩 be the set of even

𝐵 = {6,8,10} 𝐵 = {𝑥|𝑥 = 2𝑛, 𝑛 ∈ ℤ, 4 < 𝑥 < 12}
integers between 4 and 12.
 ❖ The empty set, or null set, is the set that contains no elements. The symbol ∅ or {} is used to
represent the empty set. As an example of the empty set, consider the set of natural numbers that
are negative integers.
❖ A set is finite if the number of elements in the set is a whole number. The cardinal number of a
finite set is the number of elements in the set. The cardinal number of a finite set A is denoted by
𝑛(𝐴). For instance, if 𝐴 = {1, 4, 6, 9}, then 𝑛(𝐴) = 4.
❖ Equal Sets: Set A is equal to set B, denoted by 𝐴 = 𝐵, if and only if A and B have exactly the same
elements. For instance {𝑑, 𝑒, 𝑓} = {𝑒, 𝑓, 𝑑}.
❖ Equivalent Sets: Set A is equivalent to set B, denoted by A _ B, if and only if A and B have the same
number of elements.
Set Operations

❖ The Universal Set is the set of all elements that are under consideration.This is usually denoted by
𝑈.
❖ The complement of a set A, denoted by 𝐴’, is the set of all elements of the universal set 𝑈 that are
not elements of 𝐴.

Example:
Let 𝑈 = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Find the complements of 𝑆 and 𝑇 if:
𝑆 = {𝑥 |𝑥 < 10 and 𝑥 ∈ odd counting numbers}, and
𝑇 = {2, 4, 6, 7}.

Solution:
The elements of the universal set are 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10. 𝑇 = {1, 3, 5, 7, 9}. Excluding the
elements of 𝑇 from 𝑈 gives us 𝑇’ = {2, 4, 6, 8, 10}. Also, from 𝑈, we wish to exclude the elements of 𝑆,
which are 2, 4, 6, and 7. Therefore 𝑆’ = {1, 3, 5, 8, 9, 10}.

❖ Set A is a subset of set B, denoted by 𝐴 ⊆ 𝐵, if and only if every element of A is also an element
of B.

Subset Relationships
A ⊆A, for any set A
∅ ⊆ A, for any set A

The notation 𝐴 ⊄ 𝐵 is used to denote that A is not a subset of B. To show that A is not a subset of B, it is
necessary to find at least one element of A that is not an element of B.

Example:
Determine whether each statement is true or false.
a. {5, 10, 15, 20} ⊆ {10, 15, 20, 25, 30}
b. ℤ ⊆ ℚ
c. {2, 4, 6} ⊆ {2, 4, 6}
d. ∅ ⊆ {1, 2, 3}
Solution
a. False; 5 is not an element of the bigger set.
b. True; every integer is also a rational number
 c. True; every set is a subset of itself.
d. True; the empty set is a subset of every set.

❖ Set A is a proper subset of set B, denoted by 𝐴 ⊂ 𝐵, if every element of A is an element of B, and

𝐴 ≠ 𝐵. In the previous example c., {2,4,6} is a subset of {2,4,6} but is not a proper subset because
proper subsets do not include the set itself.

The Number of Subsets of a Set

A set with n elements has 2n subsets.
Examples
1. {1, 2, 3, 4, 5, 6} has 6 elements, so it has 26 = 64 subsets.
2. {4, 5, 6, 7, 8, ... , 15} has 12 elements, so it has 212 = 4096 subsets.
3. The empty set has 0 elements, so it has 20 = 1 subset.
4. {𝑎, 𝑏, 𝑐} has 3 elements, so it has 23 = 8 subsets. (Can you name them all?)

❖ The union of sets A and B, denoted by A ∪ B, is the set that contains all the elements that belong
to A or to B or to both. In symbols, 𝑨 ∪ 𝑩 = {𝒙|𝒙 ∈ 𝑨 𝐨𝐫 𝒙 ∈ 𝑩}.

Example:
Let 𝐴 = {1, 4, 5, 7}, 𝐵 = {2, 3, 4, 5, 6}, and 𝐶 = {3, 6, 9}. Find 𝐴 ∪ 𝐵 and 𝐴 ∪ 𝐶.

Solution:
𝐴 ∪ 𝐵 = {1, 4, 5, 7} ∪ {2, 3, 4, 5, 6} = {1, 2, 3, 4, 5, 6, 7}
𝐴 ∪ 𝐶 = {1, 4, 5, 7} ∪ {3, 6, 9} = {1, 3, 4, 5, 6, 7, 9}

❖ The intersection of sets A and B, denoted by A ∩ B, is the set of elements common to both A and
B. In symbols, 𝑨 ∩ 𝑩 = {𝒙/𝒙 ∈ 𝑨 𝐚𝐧𝐝 𝒙 ∈ 𝑩}.

Example:
Let 𝐴 = {1, 4, 5, 7}, 𝐵 _ = {2, 3, 4, 5, 6}, and 𝐶 = {3, 6, 9}. Find A ∩ B and A ∩ C.

Solution:
𝐴 ∩ 𝐵 = {1, 4, 5, 7} ∩ {2, 3, 4, 5, 6} = {4, 5}
𝐴 ∩ 𝐶 = ∅, 𝐴 and 𝐶 have no common element, then they are called disjoint sets.

❖ Two sets are disjoint if their intersection is the empty set. The sets A and C in the previous
example are disjoint.

Relations and Functions

A relation is used to describe certain properties of things. That way, certain things may be connected in
some way; this is called a relation. It is clear, that things are either related, or they are not, there is no in
between.
Recall from your Algebra class in high school that ordered pairs are defined in terms of sets, cartesian
product in terms of ordered pairs and relation in terms of cartesian product. All these terms are somehow
related, so thus, function.

A function 𝐹 from a set 𝐴 to a set 𝐵 is a relation with domain 𝐴 and co-domain 𝐵 that satisfies the following
properties:

1. For every element 𝑥 in 𝐴, there is an element 𝑦 in 𝐵 such that (𝑥, 𝑦) ∈ 𝐹.

2. For all elements 𝑥 in 𝐴 and 𝑦 and 𝑧 in 𝐵, if (𝑥, 𝑦) ∈ 𝐹 and (𝑥, 𝑧) ∈ 𝐹, then 𝑦 = 𝑧.

Example:
Let 𝐴 = {2,4,6} and 𝐵 = {1,3,5}. Which among the relations 𝑅, 𝑆 and 𝑇 defined below are functions from
𝐴 to 𝐵?

a. 𝑅 = {(2,5), (4,1), (4,3), (6,5)}

b. For all (𝑥, 𝑦) ∈ 𝐴 𝑥 𝐵, (𝑥, 𝑦) ∈ 𝑆 means 𝑦 = 𝑥 + 1
c. 𝑇 is defined by the arrow diagram

2• •1

4• •3

6• •5

Solution:
a. R is not a function because it does not satisfy property 2. Ordered pairs (4,1) and (4,3) have the
same first elements. It can also be illustrated in an arrow diagram in which you can see clearly that
there are 2 arrows coming out of 4 (one to many).
b. S is not a function because it did not satisfy property 1 since 𝐴 = {(2,3), (4,5)} as determined by
𝑦 = 𝑥 + 1. If 𝑥 is 6, then 𝑦 is 7 and 7 is not in set 𝐵, therefore 6 cannot be used as the first
component in S.
c. As shown in no. 3 problem, each element in {2,4,6} is related to some element in {1,3,5} and no
element in {2,4,6} is related to more than 1 element in {1,3,5}. (many to one)Binary Operation

The word "binary" means composed of two pieces. A binary operation is simply a rule for combining two
values to create a new value. The most widely known binary operations are addition, subtraction,
multiplication and division on various sets of numbers. Thus, the binary operation can be defined as an
operation * which is performed on a set A. (Note: * is any operation.)

Properties of Binary Operation

Closure property:
An operation * on a non-empty set A has closure property, if
a ∈ A, b ∈ A if a * b ∈ A.
Let us show that addition is a binary operation on real numbers ℝ and natural numbers ℕ. So, if we add
two operands which are natural numbers 𝑎 and 𝑏, the result will also be a natural number. The same holds
good for real numbers. Hence,

+: ℝ + ℝ → ℝ is given by (𝑎, 𝑏) → 𝑎 + 𝑏
+: ℕ + ℕ → ℕ is given by (𝑎, 𝑏) → 𝑎 + 𝑏

Let us show that multiplication is a binary operation on real numbers (R) and natural numbers (N). So, if we
multiply two operands which are natural numbers a and b, the result will also be a natural number. The
same holds good for real numbers. Hence,

𝑥: ℝ × ℝ → ℝ is given by (𝑎, 𝑏) → 𝑎 𝑥 𝑏
𝑥: ℕ × ℕ → ℕ is given by (𝑎, 𝑏) → 𝑎 𝑥 𝑏