MTH299 - Homework 1 Name (print) · MTH299 - Homework 1 Name (print): Question 1. exercise 1.10...

MTH299 - Homework 1 Name (print):

Question 1. exercise 1.10 (compute the cardinality of a handful of finite sets)

Solution. Write your answer here.

Question 2. exercise 1.20 (compute the union of two sets)

Question 3. exercise 1.23 (compute the intersection of two sets)

Question 4. exercise 1.24 (compute the union, then the intersection of two sets)

Question 5. Give a justification of the claim that N ⊂ Z.

Question 6. Give a justification of the claim that Q 6⊂ Z.

Question 7. For the following three questions, answer true or false and justify your answer with a clearlywritten explanation. Let E be the subset of R2 that is defined as

E := {(x, y) ∈ R2 : y ≤ x2}.

(i) is the point (1, 3) an element of E?

(ii) is the point (1,−5) an element of E?

(iii) is the point (0, 0) an element of E?

Question 8. Let X = {0, 1, 2, 3, 4} and Y = {0, 2, 4}.

1. How many ordered pairs are in X × Y , Y ×X and Y ×X × Y respectively?

2. List the elements of the set {(a, b, c) ∈ X × Y ×X | a < b < c}.

Question 9. For A = {x ∈ R : |x− 1| ≤ 2} and B = {y ∈ R : |y − 4| ≤ 2}, give a geometric descriptionof the points in the xy-plane belonging to A×B.

MSU 1 Due: 5/19/2016


Question 1. exercise 1.33

Solution. Write your solutions here!

Question 2. exercise 1.34 (i)

Question 3. exercise 1.34 (ii)

Question 4. exercise 1.34 (iii). Use X = {1, 2, 3, 4, . . . , 10}, A = {2, 4, 6, 8, 10},B = {1, 2, 3, 4, 5}, C = {4, 5, 6, 7, 8}.

Question 5. exercise 1.34 (iv)

Question 6. Define the set, E = {x ∈ R : x2 ∈ {1, 2, 3, 7} }.

(i) Is E ⊆ N? Justify your answer.

(ii) Is E ⊆ Z? Justify your answer.

Question 7. exercise 30.28 (i), parts (a)–(d). Please note that a function is bijective if it is both injectiveand surjective. Bijective functions always have inverses, and the inverse is the unique function, f−1(x),which satisfies both f(f−1(y)) = y for all y in the range of f (which will be the domain of f−1) as well asf−1(f(x)) = x for all x in the domain of f . You have done this in calculus– just think of ex and ln(y) or x3

and y1/3.

Question 8. Define the operation

f(p)(x) :=

∫ x

0

p(t)dt.

Does f define a function from P4 to P4? Justify your answer. Does f define a function from P3 to P4?Justify your answer. Is f an injective function from P3 to P4? Justify your answer. Is f a surjective functionfrom P3 to P4? Justify your answer.

Question 9. Assume that f(x) = ex is the natural exponential function from calculus and that you know

(i) f(x) > 0 for all x ∈ R

(ii) f(x + y) = f(x)f(y) for all x, y ∈ R

(iii) f ′(0) > 0

Use the limit definition of the derivative plus items (i), (ii), and (iii) to carefully and completely justify thefollowing claim that f ′(x) > 0 for all x ∈ R. Please assume that you and your audience are familiar withthe limit rules that you learned in MTH 132.

MSU 1 Due: 5/24/2016


Question 1. There are eight different functions f : {a, b, c} → {0, 1}. List them all. Diagrams will suffice.


Question 2. Show that the function f : R2 → R2 defined by the formula f(x, y) = ((x2 + 1)y, x3) isbijective. Then find its inverse. Carefully justify that your answer does indeed yield the inverse function.



Question 5. exercise 5.3 (iii)


(note, you will skip 5.3(v))

Question 7. exercise 5.3 (vi)

Question 8. exercise 6.2 (all)





MSU 1 Due: 5/26/2016


Question 1. Let A = {x|x ∈ R and x > 0}. The function f : A → R is defined by f(x) = x2 − 4x + 5.What is the largest codomain so that f is surjective?


Question 2. Let A = {x|x ∈ R and x ≥ 2} and B = {x|x ∈ R and x ≥ 1} and the function f : A→ B isdefined by f(x) = x2 − 4x + 5. If there exists an inverse function f−1, then find the inverse function of thefunction f and specify the domain and codomain of the inverse function. Carefully justify that your answerdoes indeed yield the inverse function.

Question 3. exercise 6.11 (v)






Question 9. Let us define the following sets:

A0 = {z ∈ Z : z = 3k, for some k ∈ Z} ,A1 = {z ∈ Z : z = 3k + 1, for some k ∈ Z} ,A2 = {z ∈ Z : z = 3k + 2, for some k ∈ Z} .

(i) Which set, if any, do each of the integers in the set {3, 5,−2, 8, 24,−19} belong to?

(ii) Does Z = A0 ∪ A1 ∪ A2. Carefully justify your answer.

(iii) Show that if a, b ∈ A0, then ab ∈ A0. Carefully justify your answer.

(iv) Show that a ∈ A0, and b ∈ A1, what can you say about a + b and ab? Are they in any of these sets,A0, A1, or A2? Carefully justify your answers.

Question 10. Is the product of two odd integers odd or even? Justify your answer.

Question 11. Consider the statements:

P :√

2 is rational. Q :2

3is rational. R :

√3 is rational.

Write each of the following statements in words and indicate whether the statement is true or false.(a) (P ∧Q)⇒ R (b) (P ∧Q)⇒ (¬R) (c) ((¬P ) ∧Q)⇒ R (d) (P ∨Q)⇒ (¬R)

MSU 1 Due: 5/31/2016













Question 11. Define the function T : P2 → P2 via the assignment

T (f)(x) := f(x + 3)− 6, for all x ∈ R.

Carefully justify that T is both an injective and surjective function.(Note, the input to T is a function, and the output of T is another function. Therefore, in order to understandwhat is the new function, T (f), you must describe what the function T (f) does to its input variables. SinceT (f) ∈ P2, we know it is a polynomial of degree at most 2, hence a function from R→ R.)

MSU 1 Due: 6/2/2016


Question 1. Exercise 10.11 (i)


Question 2. Exercise 10.11 (ii), (a)–(d)

Question 3. Exercise 10.11 (ii), (e)–(h)

Question 4. Exercise 10.11 (ii), part (i)

Question 5. Exercise 11.12 (i)

Question 6. Exercise 11.12 (ii)

Question 7. Exercise 11.12 (iii) (b), (c), (f) (you do not need to turn in the other lettered questions in(iii))

Question 8. Exercise 11.12 (iv) (a), (b)

Question 9. Exercise 11.12 (iv) (c), (d)

Question 10. LetD = {E ⊆ N : E contains a finite number of elements}.

Define a function, c : D → N, via the assignment

c(E) = |E| .

(Recall that the cardinality, |E|, is defined in the text.) Answer the questions about the function c.

(i) Let B = {x ∈ N : 107 < x < 136 and x is divisible by 7}. Evaluate c(B).

(ii) Let H = {x ∈ N : x is a multiple of 10}. Can you evaluate c(H)?

(iii) Prove that c is not an injective function.

(iv) Prove that c is a surjective function.

Question 11. Let the function, I : P3 → R, be defined via the assignment rule

I(p) :=

∫ 1

0

p(x)dx.

(i) is I an injective function? Justify your answer.

(ii) is I a surjective function? Justify your answer.

MSU 1 Due: 6/7/2016


Question 1. Define the setsA = {q ∈ P2 : q(x) ≥ 0 for all x ∈ R},

B = {p ∈ P2 : p(x) = a(x− x0)2 + b, where a, b, x0 ∈ R, and a ≥ 0, b ≥ 0}.

Prove that A = B.


Question 2. Let f : R→ Z be the integer floor function

f(x) = bxc := max{z ∈ Z : z ≤ x},

that is to say that f(x) is the unique integer which is the largest integer that is smaller than or equal to x.

(i) sketch a picture of the graph of f .

(ii) evaluate f(550.1), f(−64.7), f(π), f(−e).

(iii) prove that f is a surjective function from R→ Z.

(iv) prove that f is not an injective function from R→ Z.

Question 3. Define the set U+ = {(x1, x2) ∈ R2 : x2 ≥ 0} ⊆ R2. Define the function h : R2 → U+ via theassignment

h(x1, x2) = (x1, x21 + x22).

Prove that h is not an injective function. Prove that h is a surjective function. Is h a bijective function?Does an inverse function exist for h?

MSU 1 Due: 6/9/2016


Question 1. Define the function, f : R→ R via the assignment

f(x) = −(x− 5)2 + 9,

and define the set E asE := {x ∈ R : f(x) ≥ −5}.

Prove that the set E is bounded.


Question 2. Let Ar = (−r, r). What is⋃

r∈I Ar and⋂

r∈I Ar, where I = {1, 2, 10}? Repeat with I = {x ∈R : x > 0}.

Question 3. Define the set A = [−5, 5]⋂{x ∈ R : sin((π/2)x) = 0}. Prove that

A = {−4,−2, 0, 2, 4}.

(See p. 145-146 of the text on the method by which you show two sets are equal, especially Proposition20.12.)

Question 4. Come up with a simpler expression for the following sets

i) P =⋂∞

n=1

(5− 1

n, 5 + 1

n

).

ii) Q =⋂∞

n=1

[5− 1

n, 5 + 1

n

].

iii) R =⋃∞

n=1

[1n, 1].

Question 5. Determine which of the following sets are open:

i) A0 = (0, 1).

ii) A2 = (−10, 5) ∪ (3, 23).

iii) A4 = (−∞, 0].

If you claim a set is not open, then give an example of an element of the set which fails the requirement ofthe definition of “open”.

Question 6. Prove that the sequence {√n+ 1−

√n} is convergent.

Question 7. Define the set

E =⋃i=1,2

(2i, 2i+ 1) ⊆ R.

Prove that E is an open set.

MSU 1 Due: 6/14/2016


Question 1. Assume that E ⊆ R. Let A be the statement “E is not an open set”. Let B be the statement“E is a closed set”. Give an example of E which demonstrates that the implication

A =⇒ B

is FALSE. (This is a very important thing to keep in mind when you go on to further analysis and topologycourses!!!)


Question 2. Guess the value limn→∞ an = L, and prove that {an}n∈N converges to L in the case that

an =sin(n)

20n + 1.

Question 3. Write down the negated definition of “convergent”. That is, write down what it means whena sequence is not convergent.

Question 4. Prove that for ak = ln(k), {ak}k∈N is not a convergent sequence.(You will need to recall from calculus that ln is the natural logarithm, and it is the unique function suchthat

eln(x) = x and ln(ex) = x ∀x ∈ R.

Furthermore, you will need to use the fact that ex is an increasing function. See the supplementary materialin section 4 for a couple of things about increasing functions. )

Question 5. Write down bounded sequence, {cn}n∈N, such that {cn}n∈N is not convergent. Try to provethat it is not convergent (this is actually harder than it sounds!).

Question 6. Prove the following implication:“If x, y ∈ R and 0 < x < y, then

√(x + y) 6=

√x +√y.”

(Hint... it may be very helpful to try a proof by contradiction.)

Question 7. Use a contradiction argument to justify that⋂n∈Z

[n, n + 2] = ∅

is a true statement.

Question 8 (BONUS AND CHALLENGE!!!). Define the set E as

E =

{1,

1

2,1

3,1

4, . . .

}=⋃n∈N

{ 1

n}.

Prove that E is not open. Also prove that E is not closed.

MSU 1 Due: 6/16/2016


Question 1. Prove that P4 is a vector space over R.


Question 2. Define the set V = {x ∈ R2 : x2 = 12x1}. Sketch a picture of the set V inside of the plane,

R2. Is V a vector space over the scalar field R? Use the usual addition structure given to you by R2. Justifyyour answer!

Question 3. Prove that the function L : P4 → P4, defined via L(p)(x) = p′(x) is a linear function. (Goeither to your old calculus book or to wikipedia to look up the properties of the derivative to find the thingsyou will need to show that this is a linear function.)

Question 4. Define the function

f : N→ N, with f(n) = n2 + n + 1.

Use the following two different methods to prove that f is injective.

(i) By contradiction. Suppose it is not injective and find a contradiction. I would suggest a good first lineis “Assume that n 6= m, n,m ∈ N, and f(n) = f(m). We note that it is OK to assume, without lossof generality, that m < n.” Then continue along and find a contradiction.

(ii) By proving that f is in fact a strictly increasing function. Also prove a separate deduction that anyfunction which is strictly increasing MUST ALSO be injective. The combination of these two stepswould conclude your argument.

Question 5. f(1) = 2, f(n + 1) =√

(3 + f(n)). Prove that f(n) < 2.4 for all n ≥ 1. You may use acalculator to check what are the values of some square roots.

Question 6. Prove the following proposition: For each n ∈ N, it follows that

2n ≤ 2n+1 − 2n−1 − 1.

Question 7. Prove that there is no positive integer, x, that satisfies

2x < x2 < 3x.

(Hint... try a proof by contradiction.)

MSU 1 Due: 6/21/2016


Question 1. Define the function L : R→ R as L(x) = x2. Is L a linear function? Justify your answer!!!


Question 2. Let L : R→ R be defined via L(x) = ax + b, where a, b ∈ R. Is L a linear function? (Note!!!!Don’t confuse what you may have called the graph of a “line” with L being linear. Check the definitions oflinear map.)

Question 3. Assume that V and W are both vector spaces over R and that L : V → W is a linear map.Define the set

N = {v ∈ V : L(v) = 0} ⊆ V.

Prove that N is itself another vector space. Please state that you understand that N ⊆ V , and that youwill accordingly use definition 5.4 and proposition 5.5 instead of checking all 10 properties in definition 5.1.

Question 4. Suppose L : V → W is a linear map. Prove that L is injective if and only if

{v ∈ V : Lv = 0} = {0}.Hint: Use the fact that you know that L(v + w) = L(v) + L(w) for all v, w ∈ V .

Question 5. Define the matrix, A =

(3 10c 0

), and define the map L : R2 → R2 via matrix multiplication

as

L(x) = Ax =

(3 10c 0

)(x1

x2

).

Prove that L is injective if and only if c 6= 0.(Please utilize the results in previous HW questions to help support your justification.)

Question 6. For each of the following L, answer “yes” or “no”, and briefly justify your answer:

(i) Is L : P4 → R, with L(p) = p′(1), a linear function?

(ii) Is L : P4 → R, with L(p) = p′′(0), a linear function?

(iii) Is L : P4 → R, with L(p) =∫ 1

0p(x)dx, a linear function?

(iv) Is L : P4 → P5, with L(p)(x) =∫ x

0p(s)ds, a linear function?

(v) Is L : P4 → P5, with L(p)(x) = 7 +∫ x

0p(s)ds, a linear function?

Question 7. Assume that you already know that D(p)(x) = p′(x) and I(p)(x) =∫ x

0p(s)ds are linear

functions from P4 → P5. Use a previous question to deduce that D is not injective and that I is injective.

Question 8.

V = {p ∈ P2 : ∀x ∈ R p(x) = ax2 + bx + c, with c > 0, and a, b, c ∈ R}.Is V a vector space over R?

Question 9. Let X be a finite set with n elements. Show that X has 2n distinct subsets.

Question 10. Of the following two statements, one is true and one is false. Determine which one is which,prove why the true one is true, and give an example that shows the false one is false.

(i) ∀x ∈ R,∃y ∈ R, such that y2 ≥ 3x + 5.

(ii) ∃y ∈ R,∀x ∈ R, such that y2 ≥ 3x + 5.

MSU 1 Due: 6/23/2016


Question 1. Answer true or false and completely justify. If p is prime, then p + 2 is prime.


Question 2. Prove that if a divides b and a divides c, then a divides (b + c).

Question 3. Answer true or false, and completely justify. If a, b ∈ N and a < b, then b is divisible by a.

Question 4. Answer true or false, and completely justify. If a divides b and b divides c, then a divides c.

Question 5. Answer true or false, and completely justify. If a divides b and b divides a, then a = b.

Question 6. Answer true or false, and completely justify. If a divides (b+ c), then a divides b and a dividesc.

Question 7. Prove that if x, y ∈ Z are consecutive integers, then xy is even.

Question 8. Prove that if z ∈ Z, x ≡ 1 mod z, and y ≡ 1 mod z, then xy ≡ 1 mod z.

Question 9. Prove that there exists an integer, x, such that x 6≡ 0 mod 4, but x2 ≡ 0 mod 4.

Question 10. Prove there are no integers, x, y ∈ Z, such that x + y = 100 and gcd(x, y) = 3.

Question 11. Prove that there is no integer, x ∈ Z, such that if y ≡ 2 mod 4 then xy ≡ 1 mod 4.

Question 12. Prove that if n ∈ N, then n4 + 2n3 + n2 is divisible by 4.

Question 13. Illustrate the division algorithm for the two numbers x = 1011, y = 37.

Question 14. Show all the steps of the Euclidean Algorithm (p.200 of the text) to find the gcd of x = 525and 770.

Question 15. Show that n2 − 1 is divisible by 8 when n is an odd natural number.

Question 16 (Challenge and Bonus). An integer n > 1 has the properties that n | (35m + 26) andn | (7m + 3) for some integer m. What is n?

MSU 1 Due: 6/28/2016

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