Saturday, 25 March 2017

Bounds

Upper Bound and Bounded Above:-
Let S is a subset of R. And a∈R. Then a is called upper bound of S if x≤a for all x∈S.
And then S is called bounded above.

Lower Bound and Bounded Below:- 
Let S is a subset of R. And a∈R. Then a is called lower bound of S if a≤x for all x∈S.
And then S is called bounded below.

Least Upper Bound:-
Let S is a subset of R. And a∈R. Then a is called least upper bound of S if it is satisfied two axioms.
1) x≤a for all x∈S.
2) there are exist a Upper Bound t such that a≤t.
I.e. lub is less or equal all other upper bounds.


Greatest lower Bound:-
Let S is a subset of R. And a∈R. Then a is called least upper bound of S if it is satisfied two axioms.
1) a≤x for all x∈S.
2) there are exist a lower Bound w such that w≤a.
I.e. glb is greater or equal all other lower bounds.