The number of ways to choose r elements (without replacement) from an nelement set is. = r r!(n r)!.


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1 The first exam will be on Friday, September 23, The syllabus will be sections 0.1 through 0.4 and 0.6 in Nagpaul and Jain, and the corresponding parts of the number theory handout found on the class web site. In the following lists, pages and results on the number theory handout will be referred by preceding the number with H. Following are some of the concepts and results you should know: The cardinality of X, denoted X, is the number of elements of X. Some formulas for the cardinality of combinations of sets X and Y : 1. X Y = X + Y X Y. 2. X Y = X Y. 3. P(X) = 2 X where P(X) denotes the power set of X, that is, P(X) is the set of all subsets of X. 4. {all functions f : X Y } = Y X. The number of ways to choose r elements (without replacement) from an nelement set is ( ) n n! = r r!(n r)!. Know the Division Algorithm. Know the definition of a divides b for integers a and b (notation: a b). Know the definition of the greatest common divisor of the integers a and b (notation: gcd(a, b)). Know the Euclidean Algorithm and how to use it to compute the greatest common divisor of integers a and b. Know how to use elementary row operations to codify the calculations needed for the Euclidean algorithm into a sequence of matrix operations as done in class and illustrated on Pages H.11 and H.12. Know the definition of relatively prime integers. Know the definition of least common multiple of integers a and b (notation: [a, b]). Know the definition of prime number. Know what it means for an integer a to be congruent modulo n to another integer b (notation a b mod n). Know the definition of congruence class of a modulo n (notation [a] n ). Know the definition of the number system Z n, and how to do arithmetic in Z n : [a] n + [b] n = [a + b] n [a] n [b] n = [ab] n Know the definition of [a] n is invertible in Z n, and know the criterion of invertibility of [a] n : An element [a] n Z n is invertible (or has a multiplicative inverse) if and only if gcd(a, n) = 1, that is, if and only if a and n are relatively prime. Moreover, if r and s are integers such that ar + ns = 1, then [a] 1 n = [r] n. (Theorem (text) and Proposition H.1.4.5, Page H.38.) 1
2 Know how to use the Euclidean algorithm to compute [a] 1 n, when the inverse exists. Z n is the set of invertible elements of Z n. Z n is closed under multiplication, i.e., the product of any two elements of Z n is in Z n. If p is a prime, then Z p = Z p {[0] n }. (Corollary H.1.4.6, Page 39). Know the Chinese Remainder Theorem (Theorem 0.6.4), and how to solve simultaneous congruences. The Euler phifunction at n, denoted φ(n), is defined to be the number of positive integers less than or equal to n which are relatively prime to n. It is also true that φ(n) = Z n. Know how to compute φ(n) from the two rules discussed in class (Theorem and following paragraph): 1. If p is a prime k is any positive integer, then 2. If n and m are relatively prime, then φ(p k ) = p k p k 1. φ(nm) = φ(n)φ(m). Know how to to compute φ(n) from the prime factorization of n using the two rules listed above: If n = p k 1 1 pkr r, then φ(n) = (p k 1 1 pk ) (p k r 1 pk r 1 1 ) = n ) ) (1 1p1 (1 1pr. Know Euler s Theorem: If n 2 and a is relatively prime to n, then a φ(n) 1 mod n. Know how to use Euler s theorem to evaluate a k mod n. Know what a relation on a set X is and the various properties of R: reflexive, symmetric, antisymmetric, and transitive. Know the two special types of relations: partial order (that is R is reflexive, antisymmetric, and transitive), and equivalence relation (that is, R is reflexive, symmetric, and transitive). Know the fundamental fact about an equivalence relation. Namely, an equivalence relation on set X determines a partition of X into disjoint sets called equivalence classes. (Theorem , Page 7). Know what the equivalence classes [a] R of an equivalence relation are? What is a semigroup? What is a group? What is the difference between a semigroup and a group? Know examples of groups such as Z, Z n, Z n, cyclic groups. In particular, know what elements of these groups are and what the group operation is in each case. Know the order of each of the groups which is finite. 2
3 Know the cancelation rules in a group. For example, if ab = ac then b = c. Exponential rules in groups. For example, a m a n = a m+n, (a m ) n = a mn, and (ab) 1 = b 1 a 1. What does it mean to be a subgroup? Know the criterion to be a subgroup (Theorem 0.4.3) and how to use it to check that H is a subgroup of a group G. What is a cyclic group? What is a generator of a cyclic group? What is the condition for an element a Z n to have a multiplicative inverse? (Answer: a and n should be relatively prime integers. When a has a multiplicative inverse, know how to find it using the Euclidean algorithm. What is the order of a group (denoted G )? What is the order o(g) of an element g G? (o(g) is the smallest positive integer m such that g m = e.) If o(g) = m, then g n = e if and only if m n. Every subgroup of a cyclic group is cyclic (Theorem 0.4.5). If G is a group and a G is an element of order n, then the order of a k is o(a k ) = n/ gcd(k, n). If G and H are groups, then the order of an element (a, b) in the cartesian product G H is related to the orders of a and b by the formula o((a, b)) = lcm(o(a), o(b)) (assuming both orders are finite. If G and H are finite cyclic groups of order m, n, respectively, then the direct product G H is cyclic if and only if m and n are relatively prime. Moreover, if this happens then (a, b) is a generator of G H if and only if a and b are generators of G and H respectively. (Theorem ). If G is a cyclic group of order n, then the order of every element is a divisor of n. Moreover, given any positive divisor r of n, there are exactly φ(r) elements of order r. These elements are a jn/r where j is a positive integers less than r and relatively prime to r. The left cosets of a subgroup H are the sets ah = {ah : h H}. They are precisely the equivalence classes of the equivalence relation a H b a 1 b H. The set of left cosets of H in G is denoted G/H. Lagrange s Theorem: [G : H] = G / H. G / H. In particular, H G. That is, the number of cosets of H is exactly The order of every element of a group divides the order of the group (Theorem ). If the order of G is prime, then G is cyclic. Review Exercises Be sure that you know how to do all assigned homework exercises. The following are a few supplemental exercises similar to those already assigned as homework. These exercises are listed randomly. That is, there is no attempt to give the exercises in the order of presentation of material in the text. 3
4 1. Let X and Y be sets with cardinalities X = 4 and Y = 3. The following are 5 sets constructed from these two given sets: A = X Y B = P(X) C = P(Y ) D = {Functions f : X Y } E = {Functions f : Y X} Recall that P(X) denotes the power set of X, that is, the set of all subsets of X. Explain why this report cannot be correct. (Hint: Part (a) may be useful.) List these 5 sets in order according to their cardinality, starting with the set with the smallest number of elements and ending with the set with the largest number of elements. 2. This problem involves arithmetic modulo 16. All answers should only involve expressions of the form [a] 16, with a an integer and 0 a < 16. (a) Compute [4] 16 + [15] 16. (b) Compute [4] 16 [15] 16. (c) Compute [15] (d) List the invertible elements of Z How many elements does Z 8 contain? List them. 4. (a) Find the greatest common divisor d = (803, 154) of 803 and 154, using the Euclidean Algorithm. (b) Write d = (803, 154) in the form d = s t Solve the system of congruences: 6. Compute mod 127. x 5 (mod 25) x 23 (mod 32). 7. Compute the Euler phi function φ(n) for each of the following natural numbers n. (a) n = 221 (b) n = 6125 (c) n = 341 (d) n = Let R be the relation on P defined by arb if and only if a b. Is R reflexive?, symmetric?, antisymmetric?, transitive? Is R a partial order? Is R an equivalence relation? 9. Let R be the relation on the integers Z defined by arb if and only if a 2 = b 2. Verify that R is an equivalence relation on Z. If a Z, find the equivalence class [a] R of a. 10. Let A be the set of integers defined by A = {n Z : 8 n < 10}. Define an equivalence relation on A by the rule nrm 4 (n m). You do not need to verify that this is an equivalence relation. List all of the equivalence classes for R. 4
5 11. Let G = {1, 1, i, i} C. Recall that C is the multiplicative group of nonzero complex numbers. (a) Verify that G is a subgroup of C. (Constructing a Cayley table for G may be useful.) (b) Verify that S = {1, i} is not a subgroup of G. (c) Verify that G is cyclic and list all the generators of G. 12. (a) If G is a group with G = 21, what are the possible orders of elements of G? (b) If G = 21, and H is a subgroup of G other than G itself, explain why H must be cyclic. (c) Let K be a subgroup of a group L and let L be a subgroup of a group M. What are the possible values of L if K = 6 and M = 72? 13. Let H = 5Z in Z. Then H is a subgroup of the additive group Z. Determine whether the following cosets of H are the same. (Remember the group operation is + so cosets will be written a + H, rather than ah.) (a) 12 + H and 27 + H (b) 13 + H and 2 + H (c) H and 1 + H 14. Let G = [a] be a cyclic group with o(a) = 36. Determine all of the elements of G that have order Determine if the group Z 7 Z 9 is cyclic. 5
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