MATH 3326-1
Intro to Abstract Mathematics
Spring 2009
Second Exam Homework Assignments
Problem 1. Let f : R → R be given by f (x) = x2 + 4x + 7. Show that Im(f ) = [3, ∞).
Problem 2. Let g : [2, ∞) → [0, ∞) be defined by g(x) =
bijection.
√
x − 2. Prove that g is a
Problem 3. Let A = {1, 2, 3} and B = {4, 5}. Show that there is no injective function
h : A → B.
Problem 4. Let m, n ∈ N, and suppose X is a set with exactly m elements and Y is a set
with exactly n elements. If f : X → Y is an injection, which of the following may be true?
What if f is a surjection? A bijection?
a. n > m.
b. n = m.
c. n < m.
d. Theres no way to tell.
Problem 5. For any sets A and B, let A/B denote the set A − B. Prove that the function
f : R/{2} → R defined by f (x) = x/(x − 2) is not a bijection. Find a set Y ⊆ R so that the
function g : R/{2} → Y given by the same formula is a bijection, and find g −1 .
Problem 6. Let f : X → Y and g : Y → Z be functions.
a. Prove that if f and g are both surjective then so is g ◦ f .
b. If g ◦ f is injective, do either of f or g have to be injective? Prove your answer.
c. If g ◦ f is surjective, do either of f or g have to be surjective? Prove your answer.
Problem 7. Let f : X → Y and g : Y → Z be functions.
a. Prove that if f and g are both surjective then so is g ◦ f .
b. If g ◦ f is injective, do either of f or g have to be injective? Prove your answer.
c. If g ◦ f is surjective, do either of f or g have to be surjective? Prove your answer.
Problem 8. Let n ∈ N and suppose i, j ∈ In . Define h : In → In by


x if x 6= i and x 6= j,

h(x) = j if x = i, and


i if x = j.
Prove that h is a bijection.
Problem 9. Show that if X is finite and Y ⊆ X, then Y is finite. (Hint: Use induction.)
Problem 10. Prove that for all n ∈ N, f : In → In is an injection if and only if f is also a
surjection.
Problem 11. Use the result stated in the previous problem to prove that if X is a finite set
and f : X → X, then f is an injection if and only if it is a surjection.
Problem 12. For n ∈ N, suppose a ≡ b mod n and c ≡ d mod n.
a. Show that (a + c) ≡ (b + d) mod n.
b. Show that ac ≡ bd mod n.
Problem 13. Let R+ = {x ∈ R| x > 0}. Determine, with justification, whether or not
(R+ , ·) is a group. Here · is regular multiplication.
Problem 14. Determine, with justification, whether or not (Z, −) is a group, where −
denotes subtraction.
Problem 15. Complete following Cayley table for (D4 , ◦) and use it to answer the problems
on the next page.
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(D4 , ◦) R0 R90 R180 R270 H V
R0
R0 R90 R180 R270 H V
R90
R90
R180 R180
R270 R270
H
H
V
V
F1
F1
F2
F2
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F 1 F2
F 1 F2
F2
a. Find the inverses of each element in D4 .
2
b. Compute R90
F13 HV R270 .
c. Without proof, compute the number of symmetries of the regular pentagon. Do the
same for the regular hexagon. In general, how many symmetries of the regular n-gon
exist for n ≥ 3?
Problem 16. Construct the Cayley table for Z5 .
Problem 17. In class we noted that (Zn , ·n ) is not a group, and in general, (Z∗n , ·n ) is not a
group either. Conjecture a condition on n such that (Z∗n , ·n ) is a group, and say whether or
not you believe this condition to be of the “if and only if” variety.
Problem 18. Prove or disprove the following statement:
In a group G, if ab = ca, then b = c.
In the problems below, (G, ∗) is a group and we will write a ∗ b = ab. We will denote the
inverse of a by a−1 and the identity of G by e.
Problem 19. Suppose a ∈ G such that a2 = a. Show that a = e.
Problem 20. Show that if abc = e, then bca = e.
Problem 21. Let G be a group with an even number of elements. Prove that there is an
element a 6= e such that a = a−1 .
Problem 22. Let a ∈ G and define a0 = e. Show that for n ∈ N, (a−1 )n = (an )−1 .
(Hint: Use induction and the fact that for n ∈ Z, an = an−1 a.)
Problem 23. Suppose that X ⊆ Z with X nonempty. Show that if X is bounded below,
then X has a least element.
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Problem 24. Define B(S) to denote the set of all bijections from s to itself. Show that
L = {g ∈ B(R) | g(x) = ax + b with a ∈ Q∗ , b ∈ Q} is a subgroup of (B(R), ◦). Show that if
Q∗ is replaced with Z, then this is not a group. (Hint: Be aware of the operation here when
you talk about “xy”.)
Problem 25. Suppose that H and K are subgroups of G. Show that H ∩ K is a subgroup
of G.
Problem 26. Consider the group D4 , the symmetries of the square. Find the order of each
element in D4 .
Problem 27. In class we saw that Z6 = h1i = h5i.
(a) Find all a ∈ Z7 such that hai = Z7 . Do the same for Z8 , Z9 , and Z10 .
(b) For n ∈ N, make (but do not prove) a conjecture as to when hki = Zn .
(a) Suppose that x6 = e but x2 6= e, x4 6= e, and x5 6= e. With proof, find all of the possible
values of the order of x.
(b) If y 12 = e, show that | y| 6= 7.
Problem 28. Prove that (Q, +) is not a cyclic group.
Problem 29. Use the previous problem and the result proved in class on 3/27 to show that
(R, +) and (C, +) are not cyclic groups.
Problem 30. In class we gave the following corollary: Suppose |a| = n. Then hai i = haj i
if and only if gcd(i, n) = gcd(j, n). Use that fact to prove the following two corollaries, also
given in class.
(a) Let G = hai be a cyclic group of order n. Then G = hak i if and only if gcd(n, k) = 1.
(b) For k ∈ Zn , Zn = hki if and only if gcd(n, k) = 1.
Problem 31. Consider the group U (25), which is a group under multiplication modulo 25.
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(a) List the elements of U (25).
(b) Given that U (25) = h2i, find all the generators of U (25).
Problem 32. Let G be a group and suppose that a ∈ G such that |a| = ∞. Show that
ai = aj if and only if i = j.
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