BASIC DEFINITIONS : REAL ANALYSIS #basic #definitions #real #analysis
https://youtu.be/Qz6MkcPX_FU?si=L5lR3p1Ld4bK4HnT
BASIC DEFINITIONS :
In this session , I am going to give definitions and few examples of
* set
* upper bound of a set
* lower bound of a set
* greatest lower bound or infimum of a set
* least upper bound or supremum of a set
* bounded above, bounded below and bounded sets.
* Diameter of a set.
* Distance between a point and a set
SET :
A set is a collection of numbers or elements or objects etc.
For example :
* the collection of numbers which contain 1,2,3,... is called the set of natural numbers.
* The collection of all people aged between 25-30 is called a set.
* The collection of numbers with the same properties is a set etc.
Note :
* A set which has no elements is called a Null or Empty set. Basically empty set can be
denoted by { } or Φ.
* If a set has atleast one element is called then the set is called a non-empty set.
* If a set has exactly one point then the set is called a point set.
Types of intervals :
For any real numbers a, b, (a ,b) = {x/a<x<b}
[a,b] = {x/a≤x≤b}
(a,b] = { x/a<x≤b}
[a ,b) = { x/a≤x<b}
UPPER BOUND OF A SET : An element x of an aggregate or a non-empty set S is called an upper bound of S
if a ≤ x for every a Ꜫ S. i.e. x is greater than every element of S.
For example if we consider (0,1) , every element in (0,1) is less than 1. Hence 1 is an
upper bound of (0,1). Not only 1 every number which are coming after 1 is an
upper bound of (0,1).
LEAST UPPER BOUND OR SUPREMUM :
An element x of an aggregate or a non empty set S is called a least upper bound or
supremum if
i . x is an upper bound of S
ii. If there exists an upper bound of S y such that x<y.
a least upper bound can also be written as l.u.b
In simple words " a supremum is least among all upper bounds of S."
LOWER BOUND :
An element x of an aggregate or a non-empty set S is called an lower bound of S
if a ≥ x for every a Ꜫ S. i.e. x is less than every element of S.
For example if we consider (0,1) , every element in (0,1) is greater than 0. Hence 0 is
called a lower bound of (0,1). Not only 0 every number which are coming before 0 is
a lower bound of (0,1).
GREATEST LOWER BOUND OR INFIMUM :
An element x of an aggregate or a non empty set S is called a greatest lower bound
or infimum if
i . x is a lower bound of S
ii. If there exists a lower bound of S, y such that x>y.
the greatest lower bound can also be written as g.l.b.
In simple words " an infimum is greatest among all lower bounds of S."
NOTE : * The infimum or supremum of a set may or may not lie in the set.
* In any interval (a,b), [a,b], (a,b], [a,b) the end points i.e. a and b are the infimum and supremum
respectively.
* In any interval (a,∞) , ∞ is not considered as supremum and (-∞,a), -∞ is not considered as
infimum.
BOUNDED ABOVE SET : A set S is called a bounded above set if it has at least one upper bound.
For example : (-∞,2) is bounded above as it has an upper bound 2. If a set has said
to bounded above, the set need not to have a lower bound.
BOUNDED BELOW SET : A set S is called a bounded below set if it has at least one lower bound.
For example : (2,∞) is bounded below as it has a lower bound 2. If a set has said to
bounded below, the set need not to have an upper bound.
BOUNDED SET : A set is called a bounded set if it has both lower and upper bound i.e an infimum and
a supremum.
For example (1,2) is a bounded set as it contains lower bound 1 and an upper bound
DIAMETER OF A SET :
The diameter of a non-empty set A is the greatest distance among all distances between the points of A and it is denoted by d(A).
d(A)= sup {d(x,y)/x,y in A}.
DISTANCE BETWEEN A POINT AND A SET :
Let A be a sub set of a non-empty set X and x€X. The distance between x and A is the least distance between x and the points of A i.e. inf {d(x,a)/a€A}.
#basic #definitions #real #analysis
#set #upper #bound #lower #infimum #supremum #bounded #above #below
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