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FOUNDATIONS OF ANALYSIS
SAMPLE EXAM
Directions. Please work as many problems as you can on the enclosed pages. The reader will read all of the problems and assign a numerical, grade be- tween 0 and 20, to each problem. The scores of the five problems with the highest grades will be added together to determine the examination grade.
It is important to show your work on all problems except #4. Cor- rect answers on the other problems may not receive full credit if the reasoning and computational paths to the answers are not clearly in- dicated. (The readers of the exams will not assume responsibility for finding the next steps in haphazard presentations.)
Please work without the aid of notes, books, computers, calculators, and other people.
The course instructor may be in the room at the time of the exam- ination acting as a resource. The examination is being given on an honor system.
Total
- This problem consist of four parts. A correct response for each definition will be assigned 5 points. A correct response for the proof will be assigned 9 points. The highest three scores will be summed to determine the credit earned on this problem. (One point is free.)
A. (5 pts.) Provide a mathematically precise definition of a field.
Problem #1 Continued
D. (9 pts.) Let A and B denote subsets of a set, U. Prove that
A(A โฉ B) = A โฉ Bc.
- This problem consist of four parts. A correct response for each definition will be assigned 5 points. A correct response for the proof will be assigned 9 points. The highest three scores will be summed to determine the credit earned on this problem. (One point is free.)
A. (5 pts.) Let A and B denote non-empty sets and let f : A โ B. Provide a mathematically precise definition of what it means to say that f is an onto function.
Problem #2 Continued
D. (9 pts.) Prove the following statement. A non-empty set of integers that is bounded below, contains a least element.
- This problem consist of four parts. A correct response for each definition will be assigned 5 points. A correct response for the proof will be assigned 9 points. The highest three scores will be summed to determine the credit earned on this problem. (One point is free.)
A. (5 pts.) Provide a mathematically precise definition of what it means for a sequence in R to converge to the real number, x.
Problem #3 Continued
D. (9 pts.) Let {ak}โ k=0 be a sequence in a field F. Prove that
โ^ n
k=
(ak โ akโ 1 ) = an โ a 0 , โn โ Z+.
Such a sum is called a telescoping sum.
- A correct response for each part will be assigned 2 points. No partial credit will be assigned for any one of the parts. Correct responses for ten of the eleven parts earns the maximum score of 20 points.
Determine whether each of the statements below is true or false. Enter the appropriate symbol, T or F, on the blank line at the beginning of each statement.
(1) Let A and B denote subsets of a set U. Then (A โช B)c^ = Ac^ โฉ Bc.
(2) Let S and T denote non-empty sets and let f : S โ T. Then the range of f is T.
(3) Let S and T denote non-empty sets and let f : S โ T. If the inverse function, f โ^1 , exists, then necessarily f is 1-1.
(4) Let the field F be ordered by the subset F+^ and let x, y โ F. If x > y then x โ y โ F+.
(5) A sequence of real numbers that converges to โโ is a Cauchy sequence.
(6) A real-valued function defined on a subset of R that consists of a finite number of points, is necessarily continuous.
(7) Let A and B denote non-empty sets such that AโฉB = โ
. Then the cardinality of A โช B is greater than the cardinality of A.
(8) For the integers, Z, it is true that Z ร Z and Z have the same cardinality.
(9) Let A and B denote non-empty finite sets such that A is properly contained in B. Then the cardinality of B is greater than the cardinality of A.
(10) Let S and T denote non-empty sets. Every subset of the product space, S ร T , is a function from S into T.
(11) Let a, b โ A. Whenever [a] โฉ [b] 6 = โ
, then [a] 6 = [b].
- A correct response on this problem earns a maximum score of 20 points.
Recall that the cardinality of the set of all functions from a set of cardinality n into { 0 , 1 } is 2n.
Prove the following statement. Let n โ Z+^ and let S be a set containing exactly n elements. Then the cardinality of the set of all subsets of S is 2n, i.e., | 2 S^ | = 2n.
