Upper bound of a subset
of R:-
Let S be a subset of real numbers. If there exist a real
number K , such that x≤K ⩝ x∈S
Then K is called an upper bound of set S.
If there
exists an upper bound for a set S then it is called “bounded above”.
Example:-
The set S= {………-4,-3,-2,-1} is bounded above & 9 is an upper bound.
Least upper bound or Supremum:-
If any subset S of R is bounded above
such that K is its upper bound and any real number less than K is not the upper
bound for the set S then K is called least upper bound (l.u.b.) or supremum of
S & denoted by supS.
Lower bound of a subset
of R:-
If S be
any subset of R & there exist a real number L, such that
x≥L ⩝x∈S
then L is
called the lower bound of the set S & the set is called bounded below.
Example:-
The set S= {1,2,3,4,……………..} is bounded below & 1 is called lower bound of
the set S.
Greatest lower bound or
Infimum:-
If any subset S of R is bounded below such that L is its
upper bound and any real number greater than L is not the lower bound of the
set S then L is called Greatest lower bound (g.l.b.) or infimum of the set S &
denoted by (inf.)S.
Bounded subset of real
numbers:-
A subset
S of R is said to be bounded if it is bounded above as well as bounded below
i.e. there exist supremum as well as infimum of the set S.
Thus the set S is said to be bounded if there exist two real
numbers K, L such that
L≤x≤K ⩝ x∈S
or it is
also called that S is subset closed interval [L,M].
By- Vinay Mishra
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2 comments:
You should also add some examples
Thanks Kabir Bhai for your Suggestion
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