Mathematics 230
Fall/Winter 2004-05
Homework 10.
Due Wednesday, April 6.
1. (a) Divide f (x) = x4 + x3 + x2 + x + 1 by g(x) = x + 1 with the remainder (i) over R and (ii)
over F2 .
(b) Factorize x3 + 4x2 + 3x + 5 over F7 .
(c) Find a greatest common divis
Mathematics 230
Winter 2004
Sample Final.
Problem 1. Use induction to prove that for every natural number n 1:
1
12
+
1
23
+ +
1
n(n+1)
=
n
n+1
Problem 2. We dene the following relation R on Q \ cfw_0 :
x R y xy is the square of a rational number
Prove th
Mathematics 230
Fall/Winter 2004-05
Homework 7.
Due Friday, January 28
1. (a) Compute the order of the permutation (1 3 4)(5 6) in the group S(8)
(b) Compute the order of the permutation (1 3 5)(3 4 7). (Note that this is not a product of
disjoint cycles.
Mathematics 230
Winter 2004
Sample Midterm II.
Problem 1. Let (n) be the Euler function.
1. Prove that (p) = p 1 for any prime number p.
2. Compute (72).
3. Given a public key n = 1147 and a = 17 explain how to encoded the following message:
1857305730910
Mathematics 230
Fall/Winter 2004-05
Homework 8.
Partial Solutions.
1. Find the group of symmetries of a rectangle. You may assume that the adjacent sides are not
equal, i.e. it is not a square. Describe each symmetry and complete the table of multiplicati
Mathematics 230
Fall/Winter 2004-05
Homework 10.
Partial Solutions.
1. (a) Divide f (x) = x4 + x3 + x2 + x + 1 by g(x) = x + 1 with the remainder (i) over R and (ii)
over F2 .
1
Answer: In both cases the result is f (x) = x3 + x + x+1 .
g(x)
(b) Factorize
Mathematics 230
Fall/Winter 2004-05
Homework 3.
Partial Solutions.
1. Textbook, #2.8 (page 34)
Solution.
(a) Since x2 = x for all x, we have (x + x)2 = (x + x). On the other hand, (x + x)2 =
x2 + x2 + x2 + x2 = x + x + x + x. Thus we have x + x = x + x +
Mathematics 230
Fall/Winter 2004-05
Homework 2.
Solutions to selected problems.
1. Textbook, #1.19 (page 24)
Solution: Let S be the collection of the elements a1 , a2 , . . . of the sequence. By the Well Ordering
Principle, S has a least element. Say, ak
Mathematics 230
Fall/Winter 2004-05
Homework 4.
Partial Solutions.
1. Let I be an ideal of a ring R. Prove that M2 (I), i.e. the set of 2 2 matrices with elements in I
is an ideal of M2 (R), and that
M2 (R)/M2 (I) M2 (R/I).
(1)
=
Solution. By the Ideal Te
Mathematics 230
Fall/Winter 2004-05
Homework 5.
Partial Solutions.
1. Prove that a 2 2 matrix is a unit in the ring M2 (R) if and only if its determinant is not zero.
Recall that the determinant of a 2 2 matrix is dened as
det
a b
c d
= ad bc.
Solution: A
Mathematics 230
Fall/Winter 2004-05
Homework 1. Solutions to selected problems. 1. Textbook, #1.9 (page 23) Solution: (a) This operation is reexive because x divides x for any positive integer x. It is reexive since if x divides y and y divides z, then x
Mathematics 230
Fall/Winter 2004-05
Homework 6.
Partial Solutions.
1. Prove that the set of all sequences of 0s and 1s is uncountable. (For example (0, 1, 0, 1, . . . ) is an
element of such a set.)
Solution: Suppose that the set S of all sequences is cou
Mathematics 230
Fall/Winter 2004-05
Homework 9.
Partial Solutions.
1. Let C be a code given by the formula w wwxx, where w F 3 is a word, x equals one if wt(w)
is odd and zero otherwise. Write down all the codewords in C. How many words in F 8 are not
cod
Mathematics 230
Fall/Winter 2004-05
Homework 3.
Due Friday, October 22.
1. Textbook, #2.1 (page 33)
2. Textbook, #2.3 (page 33)
3. Textbook, #2.8 (page 34)
4. (a) Prove the set S = cfw_a +
of real numbers.
2 b | a, b Q is a ring with the usual addition an
Mathematics 230
Fall/Winter 2004-05
Homework 9.
Due Friday, March 18.
1. Let C be a code given by the formula w wwxx, where w F 3 is a word, x equals one if wt(w)
is odd and zero otherwise. Write down all the codewords in C. How many words in F 8 are not
Final Exam Preparation Sheet.
Sections are taken from the textbook.
1. Functions: Sets, relations. Injective, surjective and bijective maps. Partition and equivalence
relation. (Section 1.2.)
2. Principle of Mathematical Induction. (Section 1.3.)
3. Integ
Mathematics 230
Fall/Winter 2004-05
Homework 7.
Partial Solutions.
1. (a) Compute the order of the permutation (1 3 4)(5 6) in the group S(8). Answer: 6.
(b) Compute the order of the permutation (1 3 5)(3 4 7). (Note that this is not a product of
disjoint
Material covered by Exam 2.
Sections are taken from the textbook.
1. Fields: Field of fractions of an integral domain, ideals, nite elds. (Sections 2.4.1 and 2.4.2.)
2. The number systems: General procedure how to construct complex numbers from the natura
Mathematics 230
Fall 2004
Sample Midterm I.
Problem 1.
1. Find gcd(240, 252).
2. Prove that if a prime number p divides a b, where a and b are positive integers, then either p
divides a, or p divides b.
Problem 2. Complete the following sentences:
1. A ri
Mathematics 230
Fall/Winter 2004-05
Homework 6.
Due Friday, January 14.
1. Prove that the Cartesian product of two countable sets is countable.
2. Prove that the set of all sequences of 0s and 1s is uncountable. (For example (0, 1, 0, 1, . . . ) is an
ele
Mathematics 230
Fall/Winter 2004-05
Homework 5.
Due Friday, December 3.
1. Textbook, #2.23 (page 58).
2. Prove that a 2 2 matrix is a unit in the ring M2 (R) if and only if its determinant is not zero.
Recall that the determinant of a 2 2 matrix is dened
Mathematics 230
Fall/Winter 2004-05
Homework 4.
Due Friday, Novermber 5.
1. Let I be an ideal of a ring R. Prove that M2 (I), i.e. the set of 2 2 matrices with elements in I
is an ideal of M2 (R), and that M2 (R)/M2 (I) M2 (R/I).
=
2. Textbook, #2.18 (pag
Mathematics 230
Fall/Winter 2004-05
Homework 2.
Due Friday, October 8.
1. Textbook, #1.15 (page 23)
2. Textbook, #1.19 (page 24)
3. Textbook, #1.23 (page 24)
4. (a) Prove that 507 and 391 are relatively prime (i.e. the only positive integer that divides b
Mathematics 230
Fall/Winter 2004-05
Homework 8.
Due Friday, March 4.
1. Find the group of symmetries of a rectangle. You may assume that the adjacent sides are not
equal, i.e. it is not a square. Describe each symmetry and complete the table of multiplica
Mathematics 230
Fall/Winter 2004-05
Homework 1.
Due Friday, September 24
1. Texbook, #1.3 (page 22)
2. Texbook, #1.9 (page 23)
3. Texbook, #1.14 (page 23)
4. Let A = cfw_1, 2, 3, 4 and B = cfw_1, 2, 3, 4, 5. Give an example of a function F : A B which is
a
yt zT xT V k T x kyyT kr T z y
GqhXueq XXzuewiVGmT Y GqiX z bG h y6`G'x Xh6U G
YTtf ytTp
ueIeG6 c Tk XExXkdmyQ uw igGwDnGe0Eb2qhY h #yXkueWG2DiIqYqYp6iTquee`ssP A 5HE60%7`1
w V z p xh k hc V n Y VT zT YT k kr n ky xYTtf kc C1 "
y k
Drwn
k z th Vh kr n