MATHEMATICS IN MODERN WORLD
MODULE 2
CHARACTERISTICS OF MATHEMATICAL
LANGUAGE
Importance of language - Language facilitates
communication and clarifies meaning. It allows
people to express themselves and maintains
their identity. Likewise, language bridges the
gap among people from varying origins and
culture without prejudice to their background
and upbringing.
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.
Sentence - is the analogue of an English
sentence; it is a correct arrangement of
mathematical symbols that states a complete
thought.
Characteristics of language of mathematics:
• Mathematical Language is Precise –
Mathematics can able to make very fine
distinctions or definitions among a set of
mathematical symbols.
• Mathematical Language is Concise –
Mathematicians can express otherwise
expositions or sentences briefly using the
language of mathematics.
• Mathematical Language is Powerful – One can
express complex thoughts with relative ease.
EXPRESSION VS. SENTENCES
The mathematical analogue of a ‘noun’ (like
people, place, and things) will be called an
expression. Thus, an expression is a name given
to a mathematical object of interest. The
mathematical analogue of a ‘sentence’ will also
be called a ‘sentence’. A mathematical
sentence, just as an English sentence, must
state a complete thought.
In the Mathematics language, expressions are
nouns. An expression is any number, variable or
a combination of the two separated by an
operation. Sentences in the mathematics
language are equations which are either true or
false but not both.
Synonyms: Different Names for the Same
Object
Numbers have lots of different names. This idea
is extremely important in mathematics. This
‘same object, different name’ idea plays a much
more role in math than in English.
For Example: the expressions
Some Difficulties in Math Language
• The word “is” could mean equality, inequality,
or membership in a set.
• Different uses of a number; to express
quantity (cardinal), to indicate the order
(ordinal), and as a label (nominal).
10, 6 + 4, 20 ÷ 2, (11 − 3) + 2, 2 + 2 + 2 + 2 + 2
• Mathematical objects may be represented in
many ways, such as sets and functions.
all look different, but are all just different
names for the same number.
• The words “and” & “or” means different from
its English use.
Ideas Regarding Sentences
English sentences have verbs, so do
mathematical sentences. In the mathematical
sentence, ‘3 + 4 = 7’, the verb is ‘=’. If you read
the sentence as ‘three plus four is equal to
seven’ then it’s easy to ‘hear’ the verb. Indeed,
the equal sign ‘=’, is one of the most popular
mathematical verbs.
• Sentences can be true or false. The notion of
truth (i.e. the property of being true or false) is
of fundamental importance in the mathematical
language.
FOUR BASIC CONCEPTS
1. SETS
CONVENTIONS IN MATHEMATICAL LANGUAGE
In the English language, it is conventional to use
capital letters for proper names. This
convention helps to distinguish between a
common name and a proper name.
Mathematical language also has its conventions
which help learners to distinguish between
different types of mathematical expressions.
•It is a well-defined collection of objects or
things
• The objects are called the elements /
members of the set.
• Well-defined - possible to determine
whether or not an object belongs to a given
set
• Capital letters of the English alphabet are
used to represent sets.
Notation:
Example:
𝒂 ∈ 𝑺 - “𝑎 is an element of a set 𝑆”
1. The use of commas
𝒂 ∉ 𝑺 - “𝑎 is not an element of a set 𝑆”
2. Simplification of expressions
3. The use of symbols as representation
Remarks
4. Rounding off of numbers (unless otherwise
stated)
• A set which contains no element is called
the empty or null set.
5. The use of bars in repeating decimals
• We denote the empty set by { } or ∅.
6. Graphical representation of data
• The set {∅} is not empty since it contains
one element, the empty set.
7. Proper writing of equation when introducing
new variable
8. Proof statements like: (If-Then, Thus, So,
Therefore, It follows that, Hence)
9. Omitting repeating expressions (the use of
“which equals” or “which is equal to”)
Two ways to represent sets
1. Roster method (Tabular method) –
when the elements of the set are
enumerated and separated by a comma
and enclosed in a pair of braces.
Example: 𝐴 = {𝑎, 𝑒, 𝑖, 𝑜, 𝑢}
2. Rule method (Set builder notation) – is
a method of describing a set by
enclosing within braces a descriptive
phrase and agreeing that those
elements and only those which have
the described property are objects or
elements of the set.
A set whose elements are unlimited or
uncountable, and the last element cannot
be identified.
Example: 𝐴 = {𝑥|𝑥 𝑖𝑠 𝑎 𝑐𝑜𝑙𝑙𝑒𝑐𝑡𝑖𝑜𝑛 𝑜𝑓
𝑣𝑜𝑤𝑒𝑙𝑠 𝑖𝑛 𝑡ℎ𝑒 𝐸𝑛𝑔𝑙𝑖𝑠ℎ 𝑎𝑙𝑝ℎ𝑎𝑏𝑒𝑡}
Denoted by the symbol U, is a set of all
elements under consideration.
Kinds of Sets
1. Equal Sets
Sets A and B are equal, written 𝑨 = 𝑩, if
they have the same elements.
Example: 𝐵 = {𝑥|𝑥 𝑖𝑠 𝑎 𝑠𝑒𝑡 𝑜𝑓 𝑤ℎ𝑜𝑙𝑒
𝑛𝑢𝑚𝑏𝑒𝑟𝑠}
5. Universal Set
Example: U = {𝑥|𝑥 𝑖𝑠 𝑎𝑛 𝑎𝑛𝑖𝑚𝑎𝑙 𝑖𝑛 𝑀𝑎𝑛𝑖𝑙𝑎
𝑍𝑜𝑜}
6. Joint Set
Sets that have common elements.
Example:
Example:
Given: If 𝐴 = {1, 2, 5, 9} and 𝐵 = {3, 4, 5, 8},
Given: If 𝐴 = {1, 4, 7, 9} and 𝐵 = {9, 1, 7, 4},
then Sets 𝐴 and 𝐵 are joint sets, since 5 is
common to both Sets 𝐴 and 𝐵.
then 𝑨 = 𝑩 since Sets A and B contains the
same elements.
7. Disjoint Set
2. Equivalent Sets
Sets that have no common elements.
Sets A and B are equivalent, written 𝑨 ∼ 𝑩,
if they have the same number of elements
Example:
Example:
Given: If 𝐴 = {𝑎, 𝑏, 𝑐, 𝑑} and 𝐵 = {1, 3, 5, 7,
9},
Given: If 𝐴 = {𝑎, 𝑒, 𝑖, 𝑜, 𝑢} and 𝐵 = {2, 4, 6,
8, 9},
then Sets 𝐴 and 𝐵 are disjoint sets since no
element is common.
then 𝑨 ∼ 𝑩 since Sets A and B contains the
same number of elements.
8. Subset
3. Finite Set
A set whose elements are limited or
countable, and the last element can be
identified.
Example: 𝐴 = {𝑥|𝑥 𝑖𝑠 𝑎 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟
𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 10}
4. Infinite Set
If 𝐴 and 𝐵 are sets, 𝐴 is called subset of 𝐵, if
and only if, every element of 𝐴 is also an
element of 𝐵.
Example:
Suppose If A = {𝑐, 𝑑, 𝑒} and B = {𝑎, 𝑏, 𝑐, 𝑑,
𝑒},
then 𝐴 ⊆ 𝐵, since all elements of 𝐴 is in 𝐵.
9. Power set
Given a set 𝑆 from universal set 𝑈, the
power set of 𝑆 denoted by 𝑃(𝑆), is the
collection (or sets) of all subsets of 𝑆.
Example:
Determine the power set of the following:
Example: E = {𝑎, 𝑒, 𝑖, 𝑜, 𝑢}, the cardinal
number of E is 5 or 𝑛(𝐸) = 5.
Operations on Sets
a. 𝑨 = {𝒆, 𝒇} 𝑃(𝐴) = {{𝑒},{𝑓},{𝑒, 𝑓}, ∅}
b. 𝑩 = {𝟏, 𝟐, 𝟑} 𝑃(𝐵) = {{1},{2},{3},{1, 2},{1,
3},{2, 3},{1, 2, 3}, ∅}
Other Terminologies:
1. Unit set (Singleton) - a set with only one
element.
Example: 𝐶 = {𝑥|𝑥 𝑖𝑠 𝑎 𝑤ℎ𝑜𝑙𝑒 𝑛𝑢𝑚𝑏𝑒𝑟
𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 1 𝑏𝑢𝑡 𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 3}
2. Empty Set (Null Set) – denoted by ∅ or {
}, is a unique set with no element.
Example: 𝐷 = {𝑥|𝑥 𝑖𝑠 𝑎𝑛 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛
2 𝑏𝑢𝑡 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 1}
3. Cardinal Number – of a set is the
number of elements or members in the
set. The cardinality of set 𝐴 is denoted
by 𝑛(𝐴).
4. FUNCTIONS
2. RELATION
• A relation is a rule that relates values from
a set of values (called the domain) to a
second set of values (called the range).
• The elements of the domain can be
imagined as input to a machine that applies
a rule to these inputs to generate one or
more outputs.
• A relation is also a set of ordered pairs (x,
y).
• A function is a relation where each
element in the domain is related to only
one value in the range by some rule.
• The elements of the domain can be
imagined as input to a machine that applies
a rule so that input corresponds to only one
output.
• A function is a set of ordered pairs (𝑥, 𝑦)
such that no two ordered pairs have the
same 𝑥−value but different 𝑦 −values.
Example:
• A function can be represented by the
equation 𝑦 = 𝑓 (𝑥) where 𝑦 is the dependent
variable and 𝑥 is the independent variable.
Let A and B be sets.
Classes of function
A relation 𝑹 from A to B is a subset of 𝐴 × 𝐵.
Given an ordered pair (𝑎, 𝑏) in 𝐴 × 𝐵, 𝒂 is
related to 𝒃 by 𝑹, written 𝑎 𝑅 𝑏 , if and only
if, (𝑎, 𝑏) is in 𝑅.
The set 𝐴 is called the domain of 𝑅 and the
set 𝐵 is called its range of 𝑅.
The notation for a relation 𝑅 may be written
symbolically as follows: 𝑎 𝑅 𝑏 means that
(𝑎, 𝑏) ∈ 𝑅 “𝑎 is related to 𝑏”
The notation 𝑎 𝑅 𝑏 means that 𝑎 is not
related to 𝑏 by 𝑅: 𝑎 𝑅 𝑏 means that (𝑎, 𝑏) ∉
𝑅 “𝑎 is not related to 𝑏”
3. Identity property
There exists an element 𝑒 in 𝐺, such that 𝒂
∗ 𝒆 = 𝒆 ∗ 𝒂, for all 𝒂 ∈ 𝑮.
Remark: • An identity element is unique.
That is, it is the same for all element of a
set.
4. Inverse property
For each 𝑎 ∈ 𝐺 there is an element 𝑎 −1 of
𝐺, such that 𝒂 ∗ 𝒂 −𝟏 = 𝒂 −𝟏 ∗ 𝒂 = 𝒆.
4. BINARY OPERATIONS
Let 𝐺 be a set. A binary operation on a set 𝐺 is a
function that assigns each ordered pair (𝑎, 𝑏) of
elements of 𝐺. Symbolically, 𝑎 ∗ 𝑏 ∈ 𝐺 for all 𝑎,
𝑏, 𝑐 ∈ 𝐺.
A group is a set of elements, with one
operation, that satisfies the following
properties:
(i)
(ii)
(iii)
(iv)
the set is closed with respect to the
operation,
the operation satisfies the
associative property,
there is an identity element, and
each element has an inverse.
In other word, a group is an ordered pair (𝐺, ∗)
where 𝐺 is a set and ∗ is a binary operation on 𝐺
satisfying the four properties.
Remarks:
• An inverse element is not unique in a set
but it is unique for each element.
• The inverse of a is denoted by 𝑎 −1
.
Example:
Determine whether the set of all nonnegative integers under addition is a group.
Solution:
We will apply the four properties to test the
set of all non-negative integers under
addition is a group.
Step1:
To test for closure property, we choose any
two positive integers,
8 + 4 = 12 and 5 + 10 = 15
The sum is always a number of the set.
Thus, it is closed.
1. Closure property
Step2:
If any two elements are combined using the
operation, the result must be an element of
the set. 𝒂 ∗ 𝒃 = 𝒄 ∈ 𝑮, for all 𝒂, 𝒃, 𝒄 ∈ �
To test for associative property, we choose
any three positive integers,
2. Associative property
3 + (2 + 4) = 3 + 6 = 9 and (3 + 2) + 4 = 5 + 4
=9
An operation on a set 𝐺 is associative if (and
only if) (𝒂 ∗ 𝒃) ∗ 𝒄 = 𝒂 ∗ (𝒃 ∗ 𝒄 ), for all 𝒂, 𝒃,
𝒄 ∈ 𝑮.
Thus, it also satisfies the associative
property.
Step3:
To test for identity property, we choose any
positive integer,
8 + 0 = 8; 10 + 0 = 10; 15 + 0 = 15
Thus, it also satisfies the identity property.
Step4:
To test for inverse property, we choose any
positive integer,
3 + (−3) = 0; 10 + (−10) = 0; 25 + (−25) = 0
Thus, it also satisfies the associative
property.
Thus, the set of all non-negative integers
under addition is a group, since it satisfies
the four properties.