Midterm Test

Franz-Viktor Kuhlmann 210 McLean Hall, Tel. 6111
Abstract Algebra — Math 363.3 (02) T2, 2004-2005
Midterm Test (Friday, February 25, 2005, in class)
Time: 50 minutes
This is a “closed book” examination.
Show all your work to receive full credit!
The maximum number of marks is 100. The bonus marks are counted as
long as your total does not exceed 100.
Be efficient by working first on easier problems or on bonus problems if you
cannot instantly solve a certain problem.
1) a) [8 marks] Determine the truth table for
Is this a tautology?
A ⇒ ¬(A ∧ B) .
b) [6 marks] Under the assumption that B ⊂ A, find the simplest expression
which denotes the same set:
(i) A ∩ B (ii) B \ A (iii) A ∪ B ∪ C
c) [3 bonus marks] Draw the Venn diagram for the set A ∩ (B ∪ C).
d) [4 bonus marks] Find equivalent expressions without negations:
(i) ¬∃x∀y¬∃z¬∀w ¬P (x, y, z, w) (ii) ¬(B ⇒ ¬A)
e) [2 bonus marks] What is the name of the law that tells us that
¬(A ∧ B) ⇔ ¬A ∨ ¬B ?
2) a) [15 marks] Which of the following relations are equivalence relations
— if one is not then say which property of equivalence relations is not
satisfied; if one is, then list the equivalence classes:
(i) x ∼ y if x − y is even (for x and y integers),
(ii) x ∼ y if x − y is odd (for x and y integers),
(iii) x ∼ y if x ≤ y (for x and y real numbers).
b) [4 bonus marks] Show that Z/nZ = {0+nZ , 1+nZ , . . . , n−1+nZ}
is a partition of Z.
/...2

3) [10 marks] Is there a bijection from a set with 57 elements to a set with
58 elements? Is there a bijection from a set with 58 elements to a set with 57
elements? Is there an invertible mapping from a set with 58 elements to a set
with 57 elements? Is there an onto mapping from a set with 58 elements to a
set with 57 elements? Is there an onto mapping from a set with 57 elements
to a set with 58 elements? Explain.
4) [4 bonus marks] Prove that in every group, the inverse of an element
is uniquely determined.
5) Let (G, ·) be a (not necessarily abelian!) group, and let a, b, c, d be
elements of G.
a) [6 marks] Prove: if ca = cb, then a = b. Why does this not hold for
the usual multiplication on Q? What modification is necessary to obtain a
group with multiplication from the set Q?
b) [4 marks] Determine the inverse of a 2 b −3 ac 5 d.
c) [3 bonus marks] Solve for x:
abcbcxd = ac .
Write the solution in the simplest possible form.
6) a) [6 marks] State the “One Step Subgroup Criterion”.
b) [12 marks] Use the One Step Subgroup Criterion to prove that in every
dihedral group Dn, the subset of rotations forms a subgroup. Show that it is
Abelian. How many elements does it have? Show that it is cyclic and name
at least one generator.
7) a) [5 marks] What is U(n)?
b) [5 marks] Determine the elements of U(9). What is the order of this
group?
c) [5 marks] Determine the Cayley table of U(9).
d) [4 marks] Is 2 an element of U(9)? If yes, determine the order of 2 in
U(n).
e) [3 marks] Determine the order of 2 in (Z/9Z, +).
f) [4 marks] Is U(9) cyclic? If yes, name a generator.
g) [3 marks] Determine the inverse of 2 in U(9).
h) [4 marks] Prove: in a cyclic group 〈a〉 of finite order n, the inverse of
the element a k is the element a n−k .
* * * THE END * * *