Arithmetic and Geometric Sequence
Thursday, October 10, 2024
2:52 PM
ARITHMETIC SEQUENCE
 An arithmetic sequence is a list of numbers with a definite
pattern. If you take any number in the sequence then
subtract it by the previous one, and the result is always the
same or constant then it is an arithmetic sequence.
Example
1 , 5 , 9 , 13, 17, 21, 25 . .
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MATHEMATICAL LANGUAGE AND SYMBOLS
Thursday, October 10, 2024
3:03 PM
The language of mathematics is the
system used by mathematicians to
communicate mathematical ideas
among themselves. This language
consists of a substrate of some
natural language (ex. English) using
technical terms and grammatical
conventions peculiar to mathematical
discourse, supplemented by a
specialized symbolic notation for
mathematical formulas.
CHARACTERISTICS OF
MATHEMATICAL LANGUAGE:
PRECISE, CONCISE AND
POWERFUL
The language of mathematics makes it
easy to express the kinds of symbols,
syntax and rules that mathematicians
like to do and is characterized by the
following:
➢ Precise (able to make very fine
distinctions) Example. The use of
mathematical symbols is only done
based on their meaning and purpose.
Like “+” means add, “-” means subtract,
“x” multiply and “÷” means divide.
➢ Concise (able to say things briefly)
Example. The long English sentence
can be shortened using mathematical
symbols. Eight plus two equals ten
which means 8 + 2 = 10.
➢ Powerful (able to express complex
thoughts with relative ease)
Example. The application of critical
thinking and problem-solving skill
requires the comprehension, analysis
and reasoning to obtain the correct
solution
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WRITING MATHEMATICAL
LANGUAGE AS AN
EXPRESSION OR SENTENCE
❖ an expression or mathematical
expression is a finite
combination of symbols that is
well formed according to rules
that depend on the context. 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
1+1+1+1
❖ The basic syntax for entering
mathematical formulas or
expressions in the system
enables you to quickly enter
expressions using 2-D notation.
The most common mistake is to
forget the parentheses “( )”. For
example, the expression:
1/(x+1) is different from 1/x+1
MATHEMATICAL LANGUAGE AND SYMBOLS .2
Thursday, October 10, 2024
3:03 PM
The language of mathematics is the
system used by mathematicians to
communicate mathematical ideas
among themselves. This language
consists of a substrate of some
natural language (ex. English) using
technical terms and grammatical
conventions peculiar to mathematical
discourse, supplemented by a
specialized symbolic notation for
mathematical formulas.
CHARACTERISTICS OF
MATHEMATICAL LANGUAGE:
PRECISE, CONCISE AND
POWERFUL
The language of mathematics makes it
easy to express the kinds of symbols,
syntax and rules that mathematicians
like to do and is characterized by the
following:
➢ Precise (able to make very fine
distinctions) Example. The use of
mathematical symbols is only done
based on their meaning and purpose.
Like “+” means add, “-” means subtract,
“x” multiply and “÷” means divide.
➢ Concise (able to say things briefly)
Example. The long English sentence
can be shortened using mathematical
symbols. Eight plus two equals ten
which means 8 + 2 = 10.
➢ Powerful (able to express complex
thoughts with relative ease)
Example. The application of critical
thinking and problem-solving skill
requires the comprehension, analysis
and reasoning to obtain the correct
solution
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WRITING MATHEMATICAL
LANGUAGE AS AN
EXPRESSION OR SENTENCE
❖ an expression or mathematical
expression is a finite
combination of symbols that is
well formed according to rules
that depend on the context. 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
1+1+1+1
❖ The basic syntax for entering
mathematical formulas or
expressions in the system
enables you to quickly enter
expressions using 2-D notation.
The most common mistake is to
forget the parentheses “( )”. For
example, the expression:
1/(x+1) is different from 1/x+1
Four Basic Concepts of Mathematics 4
Sunday, October 27, 2024
1:09 PM
1.SET
□ A set is a collection of welldefined objects that contains
no duplicates. The objects in
the set are called the
“elements” of the set. To
describe a set, we use braces
[ ] and capital letters to
represent it.
The following are examples of sets:
 The books on the shelves in a
library
 The bank accounts in the bank
 The set of natural numbers,
N = {..., -3, -2, -1, 0, 1, 2, 3, ...}
 The integer numbers Z=
{...,-3, -2, -1, 0, 1, 2, 3,...}
 The rational are numbers is
the quotients of integers
Q={p/q : p, q ∈ Z and q=0}
The three dots enumerating the
set’s elements are called ellipses
and indicate a continuous pattern. A
“finite set” contains elements that
can be counted and terminates at a
certain natural number, otherwise,
it is an “infinite set”.
Example:
Set A = { 2, 4, 6, 8, 10}
There is exactly one set, the empty set, or
null set, ∅, or { }, which has no members at
all. A set with only one member is called a
singleton or a singleton set. “singleton of a”
Specification of Sets
There are three main ways to specify a set:
List Notation/Roster Method by listing all
its members – list names of elements of a
set, separate them by commas, and enclose
them in braces:
Examples:
1. {1, 12, 45}
2. {George Washington, Bill Clinton}
3. {a, b, d, m}
4. “Three-dot abbreviation” {1, 2, . . . 1000}
Predicate Notation/Rule Method/ SetBuilder Notation – by stating a property of
its elements. It has a property that the
members of the set share (a condition or a
predicate that holds for members of this
set).
Examples:
{y: y > 0} is read as: “the sets of all y’s such
that y is greater than 0”.
The set of all even numbers less
than or equal to 10. The order in
which the elements are listed is not
relevant: i.e., the set {2, 4, 6, 8, 10}
is the same as the set {8, 4, 2, 10,
6}.
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Four Basic Concepts of Mathematics 4.2
Sunday, October 27, 2024
Universal Sets – a set that contains all the
elements considered in a particular
situation and denoted by
U.
1:09 PM
Recursive Rules – by defining a set
of rules which generates or define
its members
Examples:
1. The set E of even numbers
greater than 3.
a. 4 ∈ E
b. If x ∈ E, then x + 2 ∈ E
c. Nothing else belongs to E.
Examples:
• Suppose we list the digits only.
Then, U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
• Suppose we consider the whole numbers.
Then, U = {0, 1, 2, 3, . . .}
Subsets – a set is A called a subset of set
B if every element of A is also an element
of B, “A is a subset of B is written as A ⊆ B
Example: Subsets
1. A = {7, 9} is a subset of B = {7, 8, 9}
2. D = {8, 9, 10} is subset of G = {10, 9, 8}
Equal Sets Two sets are equal if
contain the same elements.
Examples:
1. { 3, 8, 9} = {8, 3, 9}
2. {6, 7, 7, 7, 7} = {6, 7}
3. {1, 3, 5, 7} ≠ {3, 5}
A proper subset is a subset that is not
equal to the original set, otherwise, it is an
improper subset.
A subset which contains all the elements of
the original set.
Example: Given {3, 5, 7} then the proper
subsets are { }, {5, 7}, {3, 5}, {3, 7}.
Equivalent Sets Two sets are
equivalent if they contain the same
number of elements.
Examples:
Which of the following sets are
equivalent?
{ö, ù, ø}, {£, €, ¥}, {1, 2, 3}, {a, b, c}
Solution: All of the given sets are
equivalent.
The improper subset is {3, 5, 7}
The Cardinality of the Set is the number
of distinct elements belonging to a finite
set. It is called the cardinal number of the
set A denoted by n(A) or card (A) and |A|.
For example, the cardinality of the set A =
{1, 2, 3, 4, 5, 6} is
equal to 6 because set A has six elements.
The cardinality of a
set is also known as the size of the set.
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Four Basic Concepts of Mathematics 4.3
Sunday, October 27, 2024
1:09 PM
Operation of Sets
Union is an operation for sets A and
B in which a set is formed that
consists of all the elements included
in A or B or both denoted by u as A
u B.
Example:
Given: U = {1, 2, 3, 4, 5, 6, 7, 8, 9},
A = {1, 3, 5, 7}, B ={2, 4, 6, 8} and
C={1, 2}, find the following:
a. A u B
b. A u C
c. (A u B) u {8}
Intersection is the set containing
all elements common to both A and
B, denoted by ∩.
Example:
Given U = {a, b, c, d, e} , A={c, d, e},
B={a,b,c, e} and C = {a} and D = {e}.
Find the following intersections of
sets;
a. B ∩ C
b. A ∩ C
c. (A ∩ B) ∩ D
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Complementation is an operation on a set
that must be performed about a
universal set, denoted by A’.
Example.
Give
U = {a, b, c, d, e}, A={c, d, e}, Find A’.
Functions and Relations
Wednesday, October 30, 2024
9:44 PM
CARTESIAN PRODUCTS OF SETS
If set A and set B are two sets then the
cartesian product of set A and set B is a
set of all ordered pairs (a,b), such that a
is an element of A and b is an element of
B. It is denoted by A × B.
Example:
A = {1, 2}
B = {x, y}
The cartesian product A x
B is.
A x B = { (1,x) , (1,y), (2,x),
(2,y) }
ORDERED PAIRS
An ordered pair, as its name suggests, is a
pair of elements that have specific
importance for the order of their
placements. Ordered pairs are usually used
in coordinate geometry to represent a
point on a coordinate plane. Also, they are
used to represent elements of a relation.
RELATION
A concept that defines the
connection between elements of two
sets.
A relation is any set of ordered
pairs. The set of all x-components
of ordered pairs is called the
domain of the relation and the set
of all y-components is called range.
Example:
A = {1, 2} B= {1, 2, 3}
A x B = {(1,1), (1,2), (1,3), (2,1), (2,2),
(2,3)
FUNCTIONS
• A function is a special type of
relation that assigns each
element in one set, called the
domain, to exactly one
element in another set, called the
codomain or range.
• It is frequently desirable to work
with a large class of
functions rather than with a
specific one, particularly in
the development of the theory.
Therefore, we use a
symbol such as f(x), which is read “f
of x”.
To denote a function of x We write,
= ( )
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Functions and Relations .2
Wednesday, October 30, 2024
9:44 PM
Types of Functions Based on Mapping
Onto Function
In an onto function, every codomain
element is related to the domain
element. In an onto function, no
One-to-One Function
element of codomain is left without
A one-to-one function is defined by f:
A → B such that every set A element is being mapped. For a function defined
by f: A → B, such that every element
connected to a distinct element in set
B. The one-to-one function is also called in set B has a pre-image in set A. The
onto function is also called a subjective
an injective function. Here every
element of the domain has a distinct
function.
image or co-domain element for the
given function.
Many-to-One Function
A many-to-one function is defined by
the function f: A → B, such that more
than one element of the set A is
connected to the same element in the
set B. In a many-to-one function, more
than one element has the same image.
In a many-to-one function, if there is
only a single value in the codomain that
is mapped with all the elements of the
domain, then it becomes a constant
function.
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Into Function
Into function is exactly opposite in
properties to an onto function. Here
certain elements in the co-domain do
not have any pre-image. The elements
in set B are excess and are not
connected to any elements in set A.
Functions and Relations .3
Wednesday, October 30, 2024
9:44 PM
One One and Onto Function
A function that is both a one and a onto
function is called a bijective function.
Here every element of the domain is
connected to a distinct element in the
codomain and every element of the
codomain has a pre-image. Also, in other
words every element of set A is
connected to a distinct element in set
B, and there is not a single element in
set B that has been left out.
Vertical Line Test
A vertical line test is helpful to find if
the given equation represents a
function or not. The vertical line test
states that a vertical line needs to cut
the graph of a function(equation) at
only one point to represent a function.
If the graph of the equation
represented in the coordinate axis, is
cut by the vertical line at more than
one point, then the graph is not a
function.
v
An algebraic function is helpful in
defining the various operations of
algebra. The algebraic function has a
variable, coefficient, constant term,
and various arithmetic operators such
as addition, subtraction, multiplication,
and division.
Algebraic Function
The algebraic function can also be
represented graphically. The algebraic
function is again classified into the
following functions based on their
degree:
• Linear functions
• Quadratic functions
• Cubic function • Polynomial functions
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Functions and Relations .4
Wednesday, October 30, 2024
9:44 PM
Trigonometric Function
The six basic trigonometric functions
are f(θ) = sin θ, f(θ) = cos θ, f(θ) = tan
θ, f(θ) = sec θ, f(θ) = cosec θ. Here the
domain value θ is the angle and is in
degrees or in radians. These
trigonometric functions have been
taken based on the ratio of the sides of
a right-angle triangle, and are based on
the Pythagoras theorem.
Inverse Trigonometric Function
The six basic inverse trigonometric
functions are
= sin−1( ),
= cos−1 ,
=tan−1 ,
= csc−1 ,
=sec−1 ,
=cot−1( ). T
he domain of the inverse trigonometric
function contains real number values
and its range has angles. The
trigonometric functions and the inverse
trigonometric functions are also
sometimes referred to as periodic
functions since the principal values are
repeated
Logarithmic and Exponential Functions
Logarithmic functions have been
derived from the exponential functions.
The logarithmic functions are
considered as the inverse of
exponential functions. Logarithmic
functions have a 'log' in the function
and it has a base. The logarithmic
function is of the form = log
.
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v
Modulus Function
The modulus function gives the
absolute value of the function,
irrespective of the sign of the input
domain value. The modulus function is
represented as f(x) = |x|. The input
value of 'x' can be a positive or a
negative expression. The graph of a
modulus function lies in the first and
the second quadrants since the
coordinates of the points on the graph
are of the form (x, y), (-x, y).
Rational Function
A function that is composed of two
functions and expressed in the form
of a fraction is rational. A rational
fraction is of the form f(x)/g(x), and
g(x) ≠ 0. The graphical representation
of these rational functions involves
horizontal/vertical asymptotes, and
the function does not touch the
asymptotes.
Functions and Relations .5
Wednesday, October 30, 2024
9:44 PM
Signum Function
The signum function helps us to know
the sign of the function and does not
give the numeric value or any other
values for the range. The range of the
signum function is limited to {-1, 0, 1}.
For the positive value of the domain,
the signum function answers 1, for
negative values the signum function
answers -1, and for the 0 value of a
domain, the image is 0. The signum
function has wide applications in
software programming.
Composite Function
The composite functions are of the
form of
,
,(
)( ) and are
made from the individual functions of
f(x), g(x), h(x). The composite functions
made of two functions have the range
of one function forming the domain for
another function. Let us consider a
composite function fog(x), which is
made up of two functions f(x) and g(x).
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Inverse Function
The inverse of a function f(x) is
denoted by f-1(x). For the inverse of
a function the domain and range of
the given function are changed as the
range and domain of the inverse
function. The inverse of a function
can be prominently seen in algebraic
functions and in inverse trigonometric
functions.