AREA OFSUBSETS OF R^2 : #AREA #subsets #R^2
AREA OFSUBSETS OF R2
Area of a Rectangle in R2 :
In R2, the rectangle is defined as the set {(x, y)/a≤ x ≤ b, c≤ y≤ d}.
The area is (b-a)(d-c).
Area of a bounded subset in R2 which is not a rectangle :
If we consider any bounded subset in R2
then how to find its area.
Here is a
procedure to find area of any bounded subsets of R2 which are not rectangles.
We illustrate to find area of bounded
subsets of R2 with an example.
Let
E={ (x,y)/x2+y2<2,x,y in Z}, a bounded subset in R2.
Therefore E= {
(0,0),(0,1),(1,0),(1,1),(0,-1),(-1,0),(-1,-1),(1,-1),(-1,1)}.
Clearly we have -1<x<1 and
-1<y<1.
Therefore E is bounded.
Let R be the rectangle with vertices
(-2,-2),(2,-2),(-2,2),(2,2).
Clearly R encloses the set E.
Now divide R into equal sub-rectangles (or
squares) of length and breath as ½ units.
If we observe closely, among all these
sub-rectangles, some rectangles covers E i .e some rectangles contains all the
elements of E and some rectangles do not contain no element of E and some
rectangles has at least one element of E.
Clearly we have the no .of rectangles which contain at least one element
of E are more than the no .of rectangles which can contain all the elements of E.
There fore the sum of the areas of sub-rectangles which has at least one element of E is greater than or equal to sum of all the sub-rectangles which completely contains E.
Let P be the division (partition) of the rectangle R.
Let s(P) denotes the
sum of the areas of the sub-rectangles which has entirely the points of E and
S(P) denotes the sum of areas of the sub-rectangles which has at least one point
E.
Clearly we have s(P)
< S(P).
Here P is not only
the partition of E. E is partitioned in countable times.
For each partition
there exist s(P) and S(P).
Let s={s(P)/P is a
partition of R} and S={S(P)/ p is a partition of R}
Next find sup s and
inf S.
Here sup s is called
the inner area of E and inf S is called the outer area of E.
If sup s=inf S, then
that value is called the area of E.
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