Chapter 4 - Definitions

Question 1

What is the definition for supremum?

Answer 1

Let A be a non empty subset of the real numbers.
The supremum of A is the least upper bound on A, is this exists.
SupA is a real number with two properties:
1. supA is an upper bound on A, that is for all x in A, x is less than or equal to supA
2. no number less than supA is an upper bound on A

Question 2

What is the definition for infinimum?

Answer 2

Let A be a non empty subset of the real numbers.
The infinimum of A is the greatest lower bound on A, if this exists.
InfA is a real number with two properties:
1. infA is a lower bound on A, that is for all x in A, x is greater than or equal to infA
2. no number less than infA is a lower bound on A.

Question 3

What is the definition for diameter?

Answer 3

The diameter of a bounded set A that is a subset of the real numbers is diamA=sup{absolute value of |x-y| : where x,y is in A}

Question 4

What is the definition for a dissection?

Answer 4

A dissection of a closed bounded interval [a,b] is a finite subset fancy D of [a,b] containing both a and b.
If fancy D has n+1 elements, we label these asubscript0, asubscript1,…, asubscriptn so that a = aubscript0 < asubscript1 < asubscript2<…<asubscriptn = b.
fancy D is a dissection of size n, and if the points in the dissection are regularly spaced i.e. if asubscriptj - asubscript(j-1) = (b-a)/n for all j, then fancy D is a regular dissection.

Question 5

What is the definition of a Riemann Sum?

Answer 5

Let f:[a,b] go to the real numbers be a bounded function and fancy D be a dissection of size n of [a,b].
For each j that exists in {1,2,…,n} let,
msubscriptj =inf{f(x): asubscript(j-1) <or= x <or= asubscriptj} and
Msubscriptj= sup{f(x): asubscript(j-1) <or= x <or= asubscriptj}
These numbers exist, since f is bounded.
The Lower Riemann sum, with respect to fancy D is lsubscript(fancyD)of(f) = the sum from j=1 to n of msubscriptj(asubscriptj-asubscript(j-1)).
The Upper Riemann sum, with respect to fancy D is
usubscript(fancyD)of(f)= the sum from j=1 to n of Msubscriptj(asubscriptj-asubscript(j-1)).

Question 6

What is the definition of a Riemann integral?

Answer 6

Let f: [a,b] go to the real numbers be a bounded function. Then its,
Lower Riemann integral is
l(f) = sup{lsubscript(fancyD)(f): fancy D is any dissection of [a,b]}
Upper Riemann integral is
u(f) = inf(usubscript(fancyD)(f): fancy D is any dissection of [a,b]}
f is Riemann integrable (on [a,b]) if l(f)=u(f), this common value is the Riemann integral of f (over [a,b]):= limit from b to a of f (b at top) or limit from b to a of f(x) dx.

Question 7

What is the definition for a refinement?

Answer 7

Given dissections of fancy D, fancyD’ of [a,b] we say that fancyD’ is a refinement of fancy D if fancy D is a sunset of fancyD’.
If fancyD\fancyD’ contains a k-point refinement of fancyD.
Note that fancy D is the unique 0-point refinement of itself.

Question 8

What is the axiom of completeness?

Answer 8

Every nonempty subset of the real numbers which is bounded above has a supremum in the real numbers.