MA241-FinalExam-Answers

Complete each statement on different types of mappings with the correct characteristic.

To show that two sets and have the same cardinality, one must construct an explicit _______________.
If and are finite sets, and is a proper subset of , and there is a function , then f cannot be a(n) _______________.
If a function has an inverse , then f is a(n) ________________.

Question 2

Match the following sequences with their limits.

The value of is:
The value of is:

Question 3

Suppose

. If

, which of the following statements is true?

Choose one answer.

	A.
	B.
	C.
	D.
	E.

Question 4

Suppose that

and

. What is the most one can say about

Choose one answer.

	A.
	B.
	C.
	D.
	E.

Question 5

To show that for every

, there exists an

such that

, one appeals to which of the following?

Choose one answer.

	A. The Archimedean Property
	B. The Well-Ordering Principle
	C. The completeness of the real numbers
	D. The Supremum Property
	E. The Axiom of Choice

Question 6

A positive sequence may not be bounded under which circumstance?

Choose one answer.

	A. It is Cauchy.
	B. It contains a convergent subsequence.
	C. It is convergent.
	D. It is nonincreasing.
	E. Its -tail is bounded for some natural number .

Question 7

Suppose that

is a Cauchy sequence. Then which of the following must be true?

Choose one answer.

	A. For every , there exists such that for all .
	B. For some , there exists such that for all .
	C. For every , there exists such that for all .
	D. For some , there exists such that for all .
	E. For every , there exists such that .

Question 8

Suppose that

and

are elements of a real inner product space

such that

and

. What can we say about

Choose one answer.

	A.
	B.
	C.
	D.
	E.

Question 9

Which of the following is NOT a closed set in

with respect to the Euclidean metric?

Choose one answer.

	A.
	B.
	C.
	D.
	E.

Question 10

Suppose

is a compact subset of

. Which of the following is false?

Choose one answer.

	A. There exists such that for every , .
	B. Every sequence in that converges has its limit in .
	C. Every sequence in has a convergent subsequence.
	D. is a complete metric space with the metric inherited from .
	E. If is a collection of sets whose union contains , then there is a finite subset of whose union also contains .

Question 11

Suppose

is a subset of a complete metric space

. Which of the following is most accurate?

Choose one answer.

	A. In order for every sequence in to contain a convergent subsequence, it is necessary that be compact.
	B. In order for every sequence in to contain a convergent subsequence, it is sufficient that be compact.
	C. In order for every sequence in to contain a convergent subsequence, it is necessary and sufficient that be compact.
	D. In order for every sequence in to contain a convergent subsequence, it is neither necessary nor sufficient that be compact.
	E. If is compact, then it cannot be the case that every sequence in to contain a convergent subsequence.

Question 12

A metric space

is complete under which condition?

Choose one answer.

	A. If every convergent sequence in is Cauchy
	B. If every Cauchy sequence in converges to a limit in M
	C. If every bounded sequence in has a convergent subsequence
	D. If every sequentially continuous function on continuous
	E. If every open cover of has a finite subcover

Question 13

Which of the following functions contradicts the statement that inverse images of connected sets under continuous maps are connected?

Choose one answer.

	A. .
	B. .
	C. .
	D. .
	E. .

Question 14

Suppose that

is a complete metric space, and let

. Which of the following is true?

Choose one answer.

	A. In order for to be complete, it is necessary that be continuous.
	B. In order for to be complete, it is necessary that be uniformly continuous.
	C. In order for to be complete, it is necessary that be differentiable.
	D. In order for to be complete, it is necessary that be continuously differentiable.
	E. None of the above

Question 15

Let

be a function between two metric spaces, let

be a compact subset of

, and let

be an open subset of

. Which of the following would NOT preclude the continuity of

Choose one answer.

	A. does not attain a minimum or maximum on .
	B. contains isolated points.
	C. There exists a sequence in such that does not contain a convergent subsequence.
	D. There exists a point in which is not an interior point of .
	E. is closed.

Question 16

Let be a sequence of functions with . Which of the following is true?

Choose one answer.

	A. In order for to be convergent on , it is necessary that for every subsequence , converges on compact subsets of .
	B. In order for to be convergent on , it is sufficient that for every subsequence , converges on compact subsets of .
	C. In order for to be convergent on , it is both necessary and sufficient that for every subsequence , converges on compact subsets of .
	D. In order for to be convergent on , it is neither necessary nor sufficient that for every subsequence , converges on compact subsets of .
	E. If for every subsequence , converges on compact subsets of , then it cannot be the case that to be convergent on .

Question 17

Let

be a sequence in

. Which of the following is NOT sufficient to guarantee that

is convergent?

Choose one answer.

	A. For every , there exists such that for all .
	B. There exists and such that for all , , and for all , .
	C. There exists such that for every , there exists such that for all .
	D. There exists such that for every , there exists such that for all , .
	E. There exists such that for every and there exists such that .

Question 18

Recall that a set

is convex if and only if for all

. Suppose that

is differentiable and that for some

for all

. Which of the following is true?

Choose one answer.

	A. In order to have for all , it is necessary that be convex.
	B. In order to have for all , it is sufficient that be convex.
	C. In order to have for all , it is necessary and sufficient that be convex.
	D. In order to have for all , it is neither necessary nor sufficient that be convex.
	E. If is convex, then it cannot be the case that for all .

Question 19

Suppose a set

is a dense subset of another set

and that both are subsets of a metric space

. Which of the following is true?

Choose one answer.

	A. If is closed, then .
	B. If is open, then so is .
	C. If has nonempty interior, then so does .
	D. If is open, then may have isolated points.
	E. If is countable, then so is .

Question 20

Let

be continuously differentiable and satisfying, for all

, the inequality

. Which of the following is true?

Choose one answer.

	A. If for some holds that then for all holds that .
	B.
	C. The function is strictly increasing.
	D. If is increasing on the interval then .
	E. Both options A and B are correct.

Question 21

Let

be differentiable and let

and

(in particular these limits exist). Which of the following implies the existence of

for which

Choose one answer.

	A. .
	B.
	C.
	D.
	E. None of the above.

Question 22

What does it mean to say that a function

is differentiable on its domain with derivative

Choose one answer.

	A. For all , given , there exists such that if , then .
	B. For all , there exist and such that if , then .
	C. Given , there exists such for all , if , then .
	D. There exists such that for all , there exists such that if , then .
	E. For all , there exist and such that if , then .

Question 23

Which of the following defines what it means for a sequence

of positive numbers in

to be properly divergent?

Choose one answer.

	A. For every and for every natural number , there exists such that .
	B. There exists some such that for every natural number there exists such that .
	C. For every there exists a natural number , such that for every , .
	D. For every and for every natural number , there exists such that .
	E. There exists a natural number such that for every if , then .

Question 24

Given a function

, and two subsets

, what does one mean by

Choose one answer.

	A. is an element of , but not of .
	B. There is no element such that .
	C. .
	D. There is an element such that .
	E. There are elements and such that .

Question 25

Let

be differentiable and define

by the formula

. Which of the following is the value of

Choose one answer.

	A.
	B.
	C.
	D.
	E. None of the above.

Question 26

Suppose that

is continuous on

. Let

. Which of the following may be false?

Choose one answer.

	A. is open.
	B. is bounded.
	C. is closed.
	D. is connected.
	E. is compact.

Question 27

Consider the following (false) claim and "proof;" find the incorrect step. Claim: All positive integers are equal. Proof:
1. It suffices to show that for any two positive integers, A and B, A = B.
2. Further, it suffices to show that for all N > 0, if A and B are positive integers which satisfy MAX(A, B) = N, then A = B.
3. (Base Case) If N = 1, then A and B, being positive integers, must both be 1. So A = B.
4. (Induction Step) Assume that the theorem is true for some value k. Take positive integers A and B with MAX(A, B) = k+1. Then, MAX((A-1), (B-1)) = k.
5. Therefore, by induction, (A-1) = (B-1). Consequently, A = B. Hence, A = B for all positive integers A and B by induction. Q.E.D.

Choose one answer.

	A. Step 1
	B. Step 2
	C. Step 3
	D. Step 4
	E. Step 5

Step 5 is the incorrect step. Even if A and B are positive integers, (A-1) and (B-1) may not be.

Question 28

is a continuous function, and

is compact, which of the following is true?

Choose one answer.

	A. is compact.
	B. is closed.
	C. is bounded.
	D. is connected.
	E. None of the above

Question 29

Suppose that a function

is monotone. Recall that

has the Darboux property on a set

, if and only if given any

, for any

. Let

be an interval. Which of the following is true?

Choose one answer.

	A. If has the Darboux property on , then it is necessary that be continuous.
	B. If has the Darboux property on , then it is sufficient that be continuous.
	C. If has the Darboux property on , then it is both necessary and sufficient that be continuous.
	D. If has the Darboux property on , then it is neither necessary nor sufficient that be continuous.
	E. If is continuous, then it cannot be the case that has the Darboux property on .

Question 30

Consider a sequence

. What does it mean to say that

Choose one answer.

	A. As , is the infimum of the limits of the sequences .
	B. As , the infima of the sets approach .
	C. As , the infima of the sets approach , where is the limit of the sequence.
	D. is the infimum of the sequence .
	E. is the infimum of the limits of the nondecreasing subsequences of .

For every

, there exists

such that for every

Question 31

Suppose that

is monotone increasing. Which of the following is false?

Choose one answer.

	A. for all .
	B. for all .
	C. has only jump discontinuities if it has any at all.
	D. is one-to-one.
	E. has at most countably many discontinuities.

Question 32

Which of the following is NOT a norm on the set of continuous functions on

Choose one answer.

	A. .
	B. .
	C. .
	D. .
	E. .

Question 33

Which of the following is NOT sufficient to imply that a subset

of a metric space

is open?

Choose one answer.

	A. is closed.
	B. The intersection of with any open set in is open.
	C. For every point in , there exists such that the open ball of radius with center is contained in .
	D. is the complement of a countable intersection of closed sets.
	E. is dense in .

Question 34

A set

is a perfect set if it is closed and if every element of

is an accumulation point. Which of the following is true?

Choose one answer.

	A. Every compact set is perfect.
	B. Every closed, unbounded set is perfect.
	C. Every closed set is compact, perfect, or both.
	D. Every perfect set is uncountable.
	E. Every perfect set is bounded.

Question 35

A set

is said to be relatively compact if

is compact. Given such a set

and a function,

, which of the following is false?

Choose one answer.

	A. Every sequence in has a convergent subsequence.
	B. is bounded.
	C. If is continuous, then is relatively compact
	D. If is continuous, and is an interval, then .
	E. If is continuous on and differentiable on , then is bounded on .

Question 36

Suppose

, where

is an interval. Let

be a sequence of functions which converges pointwise to

. Which of the following is true?

Choose one answer.

	A. If each is continuous, then so is .
	B. If each is uniformly continuous, then so is .
	C. If each is continuous, and is compact,then is continuous.
	D. If the convergence of to is also uniform, then is continuous.
	E. If the convergence of to is uniform and the elements of some subsequence of are continuous, then is continuous.

Question 37

Which of the following is sufficient to imply that a series

converges?

Choose one answer.

	A. .
	B. converges and for all n.
	C. , and is monotone decreasing.
	D. .
	E. The sequence , where , is Cauchy.

Question 38

Consider two subsets

. Recall that the symmetric difference of two sets

and

. What is the complement of this set in

Choose one answer.

	A.
	B.
	C.
	D.
	E.

Question 39

Let

be a sequence of continuously differentiable functions which converges uniformly on

to the function

. Which of the following may be false?

Choose one answer.

	A. is continuous.
	B. The sequence , where , converges uniformly on .
	C. .
	D. For all subsets holds that the sequence , where is the restriction of to , converges uniformly on .
	E. is integrable on any interval .

Question 40

The Rational Numbers have the same cardinality as which of the following?

Choose one answer.

	A. The Real Numbers
	B. The Irrational Numbers
	C. The Transcendental Numbers
	D. The Natural Numbers
	E. The 5th roots of unity

Question 41

Which of the following functions is NOT uniformly continuous on its domain?

Choose one answer.

	A. .
	B. .
	C. .
	D. .
	E. .

Question 42

Suppose a function

is uniformly continuous. Then, which of the following is true?

Choose one answer.

	A. For every , there exists such that if then .
	B. For every and , there exists such that if then .
	C. For every there exists an such that for some , if then .
	D. For every , there exists such that for some , if , then .
	E. For every and there exists some such that if , then .

Question 43

Consider a sequence of functions

, which are contraction mappings, i.e. for all

and for all

. Let

be an open, connected subset of

. Suppose that

uniformly on S and

pointwise on

. Which of the following is true?

Choose one answer.

	A. In order for uniformly on , it is necessary that be uniformly continuous.
	B. In order for uniformly on , it is sufficient that be uniformly continuous.
	C. In order for uniformly on , it is both necessary and sufficient that be uniformly continuous.
	D. In order for uniformly on , it is neither necessary nor sufficient that be uniformly continuous.
	E. If is uniformly continuous, then it cannot be the case that uniformly on .

Question 44

Suppose a sequence of functions

(with

for each

) converges uniformly to a function

. Then, which of the following is true?

Choose one answer.

	A. For every , there exists such that for all , for all .
	B. For every and , there exists such that for all .
	C. For every there exists some and some such that for all .
	D. For some small , there exists some and some such that for all .
	E. For every and , there exists some such that for all .

Question 45

For each of the following sets consider it with its natural notion of addition and scalar multiplication with a rational number. Which of the following is NOT a vector space over the field of rational numbers?

Choose one answer.

	A. The set of matrices with rational elements.
	B. The set of functions that are continuous on the closed interval .
	C. The set of real numbers.
	D. The set of polynomials of degree less than or equal to five with rational coefficients.
	E. The set of nonsingular matrices over the field of rational numbers

Question 46

The well-ordering principle of the natural numbers asserts which of the following?

Choose one answer.

	A. Every nonempty set of positive integers contains a smallest element.
	B. If , and are real numbers with , then .
	C. If , , and are real numbers with and , then .
	D. The natural numbers are unbounded.
	E. The rational numbers are countable.