Week 5: Binary Relations

Size: px
Start display at page:

Download "Week 5: Binary Relations"

Transcription

1 1 Binary Relations Week 5: Binary Relations The concept of relation is common in daily life and seems intuitively clear. For instance, let X be the set of all living human females and Y the set of all living human males. The wife-husband relation R can be thought as a relation from X to Y. For a lady x X and a gentleman y Y, we say that x is related to y by R if x is the wife of y, written as xry. To describe the relation R, we may list the collection of all ordered pairs (x, y) such that x is related to y by R. The collection of all such related ordered pairs is simply a subset of the product set X Y. This motivates the following definition of relations. Definition 1.1. Let X and Y be nonempty sets. A binary relation (or just relation) from X to Y is a subset R X Y. If (x, y) R, we say that x is related to y by R, denoted xry. If (x, y) R, we say that x is not related to y, denoted x Ry. For each element x X, we denote by R(x) the subset of elements of Y that are related to x, that is, R(x) = {y Y : xry} = {y Y : (x, y) R}. For each subset A X, we define R(A) = {y Y : x A such that xry} = x A R(x). When X = Y, we say that R is a binary relation on X. Since binary relations from X to Y are subsets of X Y, we can define intersection, union, and complement for binary relations. The complementary relation of a binary relation R X Y is the binary relation R X Y defined by x Ry (x, y) R. 1

2 The inverse relation of R is the binary relation R 1 Y X defined by yr 1 x (x, y) R. Example 1.1. Consider a family A with five children, Amy, Bob, Charlie, Debbie, and Eric. We abbreviate the names to their first letters so that A = {a, b, c, d, e}. (a) The brother-sister relation R bs is the set R bs = {(b, a), (b, d), (c, a), (c, d), (e, a), (e, d)}. (b) The sister-brother relation R sb is the set R sb = {(a, b), (a, c), (a, e), (d, b), (d, c), (d, e)}. (c) The brother relation R b is the set {(b, b), (b, c), (b, e), (c, b), (c, c), (c, e), (e, b), (e, c), (e, e)}. (d) The sister relation R s is the set {(a, a), (a, d), (d, a), (d, d)}. The brother-sister relation R bs is the inverse of the sister-brother relation R sb, i.e., R bs = R 1 sb. The brother or sister relation is the union of the brother relation and the sister relation, i.e., R b R s. The complementary relation of the brother or sister relation is the brother-sister or sister-brother relation, i.e., R b R s = R bs R sb. Example 1.2. (a) The graph of equation 3 + y2 2 2 = 1 2 is a binary relation on R. The graph is an ellipse. x 2 2

3 (b) The relation less than, denoted by <, is a binary relation on R defined by a < b a is less than b. As a subset of R 2 = R R, the relation is given by the set {(a, b) R 2 : a is less than b}. (c) The relation greater than or equal to is a binary relation on R defined by a b a is greater than or equal to b. As a subset of R 2, the relation is given by the set {(a, b) R 2 : a is greater than or equal to b}. (d) The divisibility relation about integers, defined by a b a divides b, is a binary relation on the set Z of integers. As a subset of Z 2, the relation is given by {(a, b) Z 2 : a is a factor of b}. Example 1.3. Any function f : X Y can be viewed as a binary relation from X to Y. The binary relation is just its graph G(f) = {(x, f(x)) : x X} X Y. Exercise 1. Let R X Y be a binary relation from X to Y. Let A, B X be subsets. (a) If A B, then R(A) R(B). (b) R(A B) = R(A) R(B). (c) R(A B) R(A) R(B). Proof. (a) For any y R(A), there is an x A such that xry. Since A B, then x B. Thus y R(B). This means that R(A) R(B). 3

4 (b) For any y R(A B), there is an x A B such that xry. If x A, then y R(A). If x B, then y R(B). In either case, y R(A) R(B). Thus R(A B) R(A) R(B). On the other hand, it follows from (a) that R(A) R(A B) and R(B) R(A B). Thus R(A) R(B) R(A B). (c) It follows from (a) that R(A B) R(A) and R(A B) R(B). Thus R(A B) R(A) R(B). Proposition 1.2. Let R 1, R 2 X Y be relations from X to Y. If R 1 (x) = R 2 (x) for all x X, then R 1 = R 2. Proof. If xr 1 y, then y R 1 (x). Since R 1 (x) = R 2 (x), we have y R 2 (x). Thus xr 2 y. A similar argument shows that if xr 2 y then xr 1 y. Therefore R 1 = R 2. 2 Representation of Relations Binary relations are the most important relations among all relations. Ternary relations, quaternary relations, and multi-factor relations can be studied by binary relations. There are two ways to represent a binary relation, one by a directed graph and the other by a matrix. Let R be a binary relation on a finite set V = {v 1, v 2,..., v n }. We may describe the relation R by drawing a directed graph as follows: For each element v i V, we draw a solid dot and name it by v i ; the dot is called a vertex. For two vertices v i and v j, if v i Rv j, we draw an arrow from v i to v j, called a directed edge. When v i = v j, the directed edge becomes a directed loop. The resulted graph is a directed graph, called the digraph of R, and is denoted by D(R). Sometimes the directed edges of a digraph may have to cross each other when drawing the digraph on a plane. However, the 4

5 intersection points of directed edges are not considered to be vertices of the digraph. The in-degree of a vertex v V is the number of vertices u such that urv, and is denoted by indeg (v). The out-degree of v is the number of vertices w such that vrw, and is denoted by outdeg (v). If R X Y is a relation from X to Y, we define outdeg (x) = R(x) for x X, indeg (y) = R 1 (y) for y Y. The digraphs of the brother-sister relation R bs and the brother or sister relation R b R s are demonstrated in the following. a e b d c a e b d c Definition 2.1. Let R X Y be a binary relation from X to Y, where X = {x 1, x 2,..., x m }, Y = {y 1, y 2,..., y n }. The matrix of the relation R is an m n matrix M R = [a ij ], whose (i, j)-entry is given by { 1 if xi Ry a ij = j 0 if x i Ry j. The matrix M R is called the Boolean matrix of R. When X = Y, the matrix M R is a square matrix. 5

6 Let A = [a ij ] and B = [b ij ] be m n Boolean matrices. If a ij b ij for all (i, j)-entries, we write A B. The matrix of the brother-sister relation R bs on the set A = {a, b, c, d, e} is the square matrix and the matrix of the brother or sister relation is the square matrix Proposition 2.2. For any digraph D(R) of a binary relation R V V on V, indeg (v) = outdeg (v) = R. v V v V If R is a binary relation from X to Y, then indeg (y) = R. Proof. Trivial. x X outdeg (x) = y Y Let R be a relation on a set X. A path of length k from x to y is a finite sequence x 0, x 1,..., x k, beginning with x 0 = x and x k = y, such that x 0 Rx 1, x 1 Rx 2,..., x k 1 Rx k. A path that begins and ends at the same vertex is called a cycle. For any fixed positive integer k, let R k X X denote the relation on X given by xr k y a path of length k from x to y. 6

7 Let R X X denote the relation on X given by xr y a path from x to y. The relation R is called the connectivity relation for R. Clearly, we have R = R R 2 R 3 = R k. The reachability relation of R is the binary relation R X X on X defined by xr y x = y or xr y. Obviously, R = I R R 2 R 3 = where I is the identity relation on X defined by xiy x = y. k=1 R k, We always assume that R 0 = I for any relation R on a set X. Example 2.1. Let X = {x 1,..., x n } and R = {(x i, x i+1 ) : i = 1,..., n 1}. Then R k = {(x i, x i+k ) : i = 1,..., n k}, 1 k n/2; R k =, k (n + 1)/2; R = {(x i, x j ) : i < j}. If R = {(x i, x i+1 ) : i = 1,..., n} with x n+1 = x 1, then R = X X = X 2. 3 Composition of Relations Definition 3.1. Let R X Y and S Y Z binary relations. The composition of R and S is a binary relation S R X Z from X to Z defined by k=0 x(s R)z y Y such that xry and ysz. When X = Y, the relation R is a binary relation on X. We have R k = R k 1 R, k 2. 7

8 Remark. Given relations R X Y and S Y Z, the composition S R of R and S is backward. However, some people use the notation R S instead of our notation S R. But this usage is inconsistent with the composition of functions. To avoid confusion and for aesthetic reason, we write S R as RS = {(x, z) X Z : y Y, xry, ysz}. Example 3.1. Let R X Y, S Y Z, where X = {x 1, x 2, x 3, x 4 }, Y = {y 1, y 2, y 3 }, Z = {z 1, z 2, z 3, z 4, x 5 }. x 1 x 2 x 3 x 4 R y y y S z z z z z x 1 x 2 x 3 x 4 RS Example 3.2. For the brother-sister relation, sister-brother relation, brother relation, and sister relation on A = {a, b, c, d, e}, we have R bs R sb = R b, R sb R bs = R s, R bs R s = R bs, R bs R bs =, R b R b = R b, R b R s =. Let X 1, X 2,..., X n, X n+1 be nonempty sets. Given relations R i X i X i+1, 1 i n. We define a relation R 1 R 2 R n X 1 X n+1 from X 1 to X n+1 by xr 1 R 2 R n y, if and only if there exists a sequence x 1, x 2,..., x n, x n+1 with x 1 = x, x n+1 = y such that x 1 R 1 x 2, x 2 R 2 x 3,..., x n R n x n+1. 8 z 1 z 2 z 3 z 4 z 5

9 Theorem 3.2. Given relations R 1 X 1 X 2, R 2 X 2 X 3, R 3 X 3 X 4. We have R 1 R 2 R 3 = R 1 (R 2 R 3 ) = (R 1 R 2 )R 3. as relations from X 1 to X 4. Proof. For x X 1, y X 4, we have xr 1 (R 2 R 3 )y x 2 X 2, xr 1 x 2, x 2 R 2 R 3 y x 2 X 2, xr 1 x 2 ; x 3 X 3, x 2 R 2 x 3, x 3 R 3 y x 2 X 2, x 3 X 3, xr 1 x 2, x 2 R 2 x 3, x 3 R 3 y xr 1 R 2 R 3 y. Similarly, x(r 1 R 2 )R 3 y xr 1 R 2 R 3 y. Proposition 3.3. Let R i X Y be relations, i = 1, 2. (a) If R W X, then R(R 1 R 2 ) = RR 1 RR 2. (b) If S Y Z, then (R 1 R 2 )S = R 1 S R 2 S. Proof. (a) For any wr(r 1 R 2 )y, x X such that wrx and x(r 1 R 2 )y. Then xr 1 y or xr 2 y. Thus wrr 1 y or wrr 2 y. Namely, w(rr 1 RR 2 )y. Conversely, for any w(rr 1 RR 2 )y, we have wrr 1 y or wrr 2 y. Then x X such that wrx and xr 1 y, or, wrx and xr 2 y. This means that wrx and either xr 1 y or xr 2 y. Thus wrx and x(r 1 R 2 )y. Namely, wr(r 1 R 2 )y. The proof for (b) is similar. Exercise 2. Let R i X Y be relations, i = 1, 2,.... (a) If R W X, then R ( i=1 R i) = i=1 RR i. (b) If S Y Z, then ( i=1 R i)s = i=1 R is. 9

10 For the convenience of representing composition of relations, we introduce the Boolean operations and on real numbers. For a, b R, define a b = min{a, b}, a b = max{a, b}. Exercise 3. For a, b, c R, a (b c) = (a b) (a c), a (b c) = (a b) (a c). Proof. We only prove the first formula. The second one is similar. Case 1: b c. If a c, then the left side is a (b c) = a c = c. The right side is (a b) (a c) = b c = c. If b a c, then the left side is a (b c) = a c = a. The right side is (a b) (a c) = b a = a. If a b c, then the left side is a (b c) = a c = a. The right side is (a b) (a c) = a a = a. Case 2: b c. If a c, then a (b c) = a c = a and (a b) (a c) = a a = a. If b a c, then a (b c) = a c = a and (a b) (a c) = a c = a. If a b, then a (b c) = a b = b and (a b) (a c) = b c = b. Sometimes it is more convenient to write the Boolean operations as a b = min{a, b}, a b = max{a, b}. For real numbers a 1, a 2,..., a n, we define n n a i = a i = max{a 1, a 2,..., a n }. i=1 i=1 For an m n matrix A = [a ij ] and an n p matrix B = [b jk ], the Boolean multiplication of A and B is an m p matrix A B = [c ik ], whose (i, k)-entry is defined by n n c ik = (a ij b jk ) = (a ij b jk ). j=1 Theorem 3.4. Let R X Y, S Y Z be relations, where X = {x 1,..., x m }, Y = {y 1,..., y n }, Z = {z 1,..., z p }. 10 j=1

11 Let M R, M S, M RS be matrices of R, S, RS respectively. Then M RS = M R M S. Proof. We write M R = [a ij ], M S = [b jk ], and M R M S = [c ik ], M RS = [d ik ]. It suffices to show that c ik = d ik for any (i, k)-entry. Case I: c ik = 1. Since c ik = n j=1 (a ij b jk ) = 1, there exists j 0 such that a ij0 b j0 k = 1. Then a ij0 = b j0 k = 1. In other words, x i Ry j0 and y j0 Sz k. Thus x i RSz k by definition of composition. Therefore d ik = 1 by definition of Boolean matrix of RS. Case II: c ik = 0. Since c ik = n j=1 (a ij b jk ) = 0, we have a ij b jk = 0 for all j. Then there is no j such that a ij = 1 and b jk = 1. In other words, there is no y j Y such that both x i Ry j and y j Sz k. Thus x i is not related to z k by definition of RS. Therefore d ik = 0. 4 Special Relations We are interested in some special relations satisfying certain properties. For instance, the less than relation on the set of real numbers satisfies the socalled transitive property: if a < b and b < c, then a < c. Definition 4.1. A binary relation R on a set X is said to be (a) reflexive if xrx for all x in X; (b) symmetric if xry implies yrx; (c) transitive if xry and yrz imply xrz. The relation R is called an equivalence relation if it is reflexive, symmetric, and transitive. And in this case, if xry, we say that x and y are equivalent. The relation I X = {(x, x) x X} is called the identity relation. The relation X 2 is called the complete relation. 11

12 Example 4.1. Many family relations are binary relations on the set of human beings. a) The strict brother relation R b : xr b y x and y are both males and have the same parents. (symmetric and transitive) b) The strict sister relation R s : xr s y x and y are both females and have the same parents. (symmetric and transitive) c) The strict brother-sister relation R bs : xr bs y x is male, y is female, x and y have the same parents. d) The strict sister-brother relation R sb : xr sb x is female, y is male, and x and y have the same parents. e) The generalized brother relation R b : xr by x and y are both males and have the same father or the same mother. (symmetric, not transitive) f) The generalized sister relation R s: xr sy x and y are both females and have the same father or the same mother. (symmetric, not transitive) g) The relation R: xry x and y have the same parents. (reflexive, symmetric, and transitive; equivalence relation) h) The relation R : xr y x and y have the same father or the same mother. (reflexive and symmetric) Example 4.2. (a) The less than relation < on the set of real numbers is a transitive relation. (b) The less than or equal to relation on the set of real numbers is a reflexive and transitive relation. (c) The divisibility relation on the set of positive integers is a reflexive and transitive relation. (d) Given a positive integer n. The congruence modulo n is a relation n on Z defined by a n b b a is a multiple of n. 12

13 The standard notation for a n b is a b mod n. The relation n is an equivalence relation on Z. Theorem 4.2. Let R be a relation on a set X. Then (a) R is reflexive I R all diagonal entries of M R are 1. (b) R is symmetric R = R 1 M R is a symmetric matrix. (c) R is transitive R 2 R M 2 R M R. Proof. (a) and (b) are trivial. (c) R is transitive R 2 R. For any (x, y) R 2, there exists z X such that (x, z) R, (z, y) R. Since R is transitive, then (x, y) R. Thus R 2 R. R 2 R R is transitive. For (x, z) R and (z, y) R, we have (x, y) R 2 R. Then (x, y) R. Thus R is transitive. Note that for any relations R and S on X, we have R S M R M S. Since M R is the matrix of R, then M 2 R = M RM R = M RR = M R 2 is the matrix of R 2. Thus R 2 R M 2 R M R. 5 Equivalence Relations and Partitions The most important binary relations are equivalence relations. We will see that an equivalence relation on a set X will partition X into disjoint equivalence classes. Example 5.1. Consider the congruence relation 3 on Z. For each a Z, define [a] = {b Z : a 3 b} = {b Z : a b mod 3}. It is clear that Z is partitioned into three disjoint subsets [0] = {0, ±3, ±6, ±9,...} = {3k : k Z}, [1] = {1, 1 ± 3, 1 ± 6, 1 ± 9,...} = {3k + 1 : k Z}, [2] = {2, 2 ± 3, 2 ± 6, 2 ± 9,...} = {3k + 2 : k Z}. 13

14 Moreover, for all k Z, [0] = [3k], [1] = [3k + 1], [2] = [3k + 2]. Theorem 5.1. Let be an equivalence relation on a set X. For each x of X, let [x] = {y X : x y}, the set of elements equivalent to x. Then (a) x [x] for any x X, (b) [x] = [y] if x y, (c) [x] [y] = if x y, (d) X = x X [x]. The subset [x] is called an equivalence class of, and x is called a representative of [x]. Proof. (a) It is trivial because is reflexive. (b) For any z [x], we have x z by definition of [x]. Since x y, we have y x by the symmetric property of. Then y x and x z imply that y z by transitivity of. Thus z [y] by definition of [y]; that is, [x] [y]. Since is symmetric, we have [y] [x]. Therefore [x] = [y]. (c) Suppose [x] [y] is not empty, say z [x] [y]. Then x z and y z. By symmetry of, we have z y. Thus x y by transitivity of, a contradiction. (d) This is obvious because x [x] for any x X. Definition 5.2. A partition of a nonempty set X is a collection of subsets of X such that (a) A i for all i; (b) A i A j = if i j; (c) X = j J A j. P = {A j : j J} 14

15 Each subset A j is called a block of the partition P. Theorem 5.3. Let P be a partition of a set X. Let R P denote the relation on X defined by xr P y a block A j P such that x, y A j. Then R P is an equivalence relation on X, called the equivalence relation induced by P. Proof. (a) For each x X, there exits one A j such that x A j. Then by definition of R P, xr P x. Hence R P is reflexive. (b) If xr P y, then there is one A j such that x, y A j. By definition of R P, yr P x. Thus R P is symmetric. (c) If xr P y and yr P z, then there exist A i and A j such that x, y A i and y, z A j. Since y A i A j and P is a partition, it forces A i = A j. Thus xr P z. Therefore R P is transitive. Given an equivalence relation R on a set X. The collection P R = {[x] : x X} of equivalence classes of R is a partition of X, called the quotient set of X modulo R. Let E(X) denote the set of all equivalence relations on X and P (X) the set of all partitions of X. Then we have functions f : E(X) P (X), f(r) = P R ; g : P (X) E(X), g(p) = R P. The functions f and g satisfy the following properties. Theorem 5.4. Let X be a nonempty set. Then for any equivalence relation R on X, and any partition P of X, we have (g f)(r) = R, (f g)(p) = P. In other words, f and g are inverse of each other. 15

16 Proof. Recall (g f)(r) = g(f(r)), (f g)(p) = f(g(p)). Then x[g(p R )]y A P R s.t. x, y A xry; A f(r P ) x X s.t. A = R P (x) A P. Thus g(f(r)) = R and f(g(p)) = P. Example 5.2. Let Z + be the set of positive integers. Define a relation on Z Z + by (a, b) (c, d) ad = bc. Is an equivalence relation? If Yes, what are the equivalence classes? Let R be a relation on a set X. The reflexive closure of R is the smallest reflexive relation r(r) on X that contains R; that is, a) R r(r), b) if R is a reflexive relation on X and R R, then r(r) R. The symmetric closure of R is the smallest symmetric relation s(r) on X such that R s(r); that is, a) R s(r), b) if R is a symmetric relation on X and R R, then s(r) R. The transitive closure of R is the smallest transitive relation t(r) on X such that R t(r); that is, a) R t(r), b) if R is a transitive relation on X and R R, then t(r) R. Obviously, the reflexive, symmetric, and transitive closures of R must be unique respectively. Theorem 5.5. Let R be a relation R on a set X. Then a) r(r) = R I; b) s(r) = R R 1 ; 16

17 c) t(r) = k=1 Rk. Proof. (a) and (b) are obvious. (c) Note that R ( ) R i R j = i=1 j=1 k=1 Rk and i,j=1 R i R j = i,j=1 R i+j = R k k=2 R k. This shows that k=1 Rk is a transitive relation, and R k=1 Rk. Since each transitive relation that contains R must contain R k for all integers k 1, we see that k=1 Rk is the transitive closure of R. Theorem 5.6. Let R be a relation on a set X with X = n 2. Then t(r) = R R 2 R n 1. In particular, if R is reflexive, then t(r) = R n 1. Proof. It is enough to show that for all k n, R k This is equivalent to showing that R k k 1 i=1 Ri for all k n. Let (x, y) R k. There exist elements x 1,..., x k 1 X such that n 1 i=1 R i. (x, x 1 ), (x 1, x 2 ),..., (x k 1, y) R. Since X = n 2 and k n, the following sequence x = x 0, x 1, x 2,..., x k 1, x k = y has k + 1 terms, which is at least n + 1. Then two of them must be equal, say, x p = x q with p < q. Thus q p 1 and k=1 (x 0, x 1 ),..., (x p 1, x p ), (x q, x q+1 ),..., (x k 1, x k ) R. Therefore (x, y) = (x 0, x k ) R k (q p) 17 k 1 i=1 R i.

18 That is R k k 1 i=1 R i. If R is reflexive, then R k R k+1 for all k 1. Hence t(r) = R n 1. Proposition 5.7. Let R be a relation on a set X. Then I t(r R 1 ) is an equivalence relation. In particular, if R is reflexive and symmetric, then t(r) is an equivalence relation. Proof. Since I t(r R 1 ) is reflexive and transitive, we only need to show that I t(r R 1 ) is symmetric. Let (x, y) I t(r R 1 ). If x = y, then obviously (y, x) I t(r R 1 ). If x y, then (x, y) t(r R 1 ). Thus (x, y) (R R 1 ) k for some k 1. Hence there is a sequence such that Since R R 1 is symmetric, we have x = x 0, x 1,..., x k = y (x i, x i+1 ) R R 1, 0 i k 1. (x i+1, x i ) R R 1, 0 i k 1. This means that (y, x) (R R 1 ) k. Hence (y, x) I t(r R 1 ). Therefore I t(r R 1 ) is symmetric. In particular, if R is reflexive and symmetric, then obviously I t(r R 1 ) = t(r). This means that t(r) is reflexive and symmetric. Since t(r) is automatically transitive, so t(r) is an equivalence relation. 18

19 Let R be a relation on a set X. The reachability relation of R is a relation R on X defined by such that That is, R = I t(r). xr y x = y or finite x 1, x 2,..., x k (x, x 1 ), (x 1, x 2 ),..., (x k, y) R. Theorem 5.8. Let R be a relation on a set X. Let M and M be the Boolean matrices of R and R respectively. If X = n, then Moreover, if R is reflexive, then Proof. It follows from Theorem 5.6. M = I M M 2 M n 1. R k R k+1, k 1; M = M n 1. 6 Washall s Algorithm Let R be a relation on X = {x 1,..., x n }. Let y 0, y 1,..., y m be a path in R. The vertices y 1,..., y m 1 are called interior vertices of the path. For each k with 0 k n, we define the Boolean matrix W k = [w ij ], where w ij = 1 if there is a path in R from x i to x j whose interior vertices are contained in X k, otherwise w ij = 0, where X 0 = and X k = {x 1,..., x k } for k 1. Since the interior vertices of any path in R is obviously contained in the whole set X = {x 1,..., x n }, the (i, j)-entry of W n is equal to 1 if there is a path in R from x i to x j. Then W n is the matrix of the transitive closure t(r) of R, that is, W n = M t(r). 19

20 Clearly, W 0 = M R. We have a sequence of Boolean matrices M R = W 0, W 1, W 2,..., W n. The so-called Warshall s algorithm is to compute W k from W k 1, k 1. Let W k 1 = [s ij ] and W k = [t ij ]. If t ij = 1, there must be a path x i = y 0, y 1,..., y m = x j from x i to x j whose interior vertices y 1,..., y m 1 are contained in {x 1,..., x k }. We may assume that y 1,..., y m 1 are distinct. If x k is not an interior vertex of this path, that is, all interior vertices are contained in {x 1,..., x k 1 }, then s ij = 1. If x k is an interior vertex of the path, say x k = y p, then there two sub-paths x i = y 0, y 1,..., y p = x k, x k = y p, y p+1,..., y m = x j whose interior vertices y 1,..., y p 1, y p+1,..., y m 1 are contained in {x 1,..., x k 1 } obviously. It follows that s ik = 1, s kj = 1. We conclude that t ij = 1 { sij = 1 or s ik = 1, s kj = 1 for some k. Theorem 6.1 (Warshall s Algorithm for Transitive Closure). Working on the Boolean matrix W k 1 to produce W k. a) If the (i, j)-entry of W k 1 is 1, so is W k. Keep 1 there. b) If the (i, j)-entry of W k 1 is 0, then check the entries of W k 1 at (i, k) and (k, j). If both entries are 1, then change the (i, j)-entry in W k 1 to 1. Otherwise, keep 0 there. Example 6.1. Consider the relation R on A = {1, 2, 3, 4, 5} given by the Boolean matrix M R =

21 By Warshall s algorithm, we have W 0 = W 1 = W 2 = W 3 = W 4 = W 5 = (3, 1), (1, 5) (no change) (2, 3), (3, 1) (2, 3), (3, 5) (5, 3), (3, 1) (5, 3), (3, 5) (no change) (1, 5), (5, 1) (1, 5), (5, 3) (1, 5), (5, 4) (2, 5), (5, 4) (3, 5), (5, 3) (3, 5), (5, 4). 21

22 The binary relation for the Boolean matrix W 5 is the transitive closure of R Definition 6.2. A binary relation R on a set X is called a) asymmetric if xry implies y Rx; b) antisymmetric if xry and yrx imply x = y. Problem Set 4 1. Let R be a binary relation from X to Y, A, B X. (a) If A B, then R(A) R(B). (b) R(A B) = R(A) R(B). (c) R(A B) R(A) R(B). Proof. (a) For any y R(A), there is an a A such that (a, y) R. Obviously, a B. Thus b R(B). (b) Since R(A) R(A B), R(B) R(A B), we have R(A) R(B) R(A B). On the other hand, for any y R(A B), there is an x A B such that (x, y) R. Then either x A or x B. Thus y R(A) or y R(B); i.e., y R(A) R(B). Therefore R(A) R(B) R(A B). (c) It follows from (a) that R(A B) R(A) and R(A B) R(B). Hence R(A B) R(A B). 22

23 2. Let R 1 and R 2 be relations from X to Y. If R 1 (x) = R 2 (x) for all x X, then R 1 = R 2. Proof. For any (x, y) R 1, we have y R 1 (x). Since R 1 (x) = R 2 (x), then y R 2 (x). Thus (x, y) R 2. Similarly, for any (x, y) R 2, we have (x, y) R 2. Hence R 1 = R Let a, b, c R. Then a (b c) = (a b) (a c), a (b c) = (a b) (a c). Proof. Note that the cases b < c and b > c are equivalent. There are three essential cases to be verified. Case 1: a < b < c. We have Case 2: b < a < c. We have Case 3: b < c < a. We have a (b c) = a = (a b) (a c), a (b c) = b = (a b) (a c). a (b c) = a = (a b) (a c), a (b c) = a = (a b) (a c). a (b c) = c = (a b) (a c), a (b c) = a = (a b) (a c). 4. Let R i X Y be a family of relations from X to Y, i I. (a) If R W X, then R ( i I R ) i = i I RR i; (b) If S Y Z, then ( i I R ) i S = i I R is. 23

24 Proof. (a) Note that (w, y) R ( i I R i) x X s.t. (w, x) R and (x, y) i I R i; and (x, y) i I R i i 0 I s.t. (x, y) R i. Then (w, y) R ( i I R i) i0 I s.t. (w, y) RR i (w, y) i I RR i. (b) Note that (x, z) ( i I R i) S y Y s.t. (x, y) R and (x, y) i I R i; and (x, y) i I R i i 0 I s.t. (x, y) R i. Then (w, y) R ( i I R i) i0 I s.t. (w, y) RR i (w, y) i I RR i. 5. Let R i (1 i 3) be relations on A = {a, b, c, d, e} whose Boolean matrices are M 1 =, M 2 =, M 3 = (a) Draw the digraphs of the relations R 1, R 2, R 3. (b) Find the Boolean matrices for the relations and verify that R 1 1, R 2 R 3, R 1 R 1, R 1 R 1 1, R 1 1 R 1; R 1 R 1 1 = R 2, R 1 1 R 1 = R 3. (c) Verify that R 2 R 3 is an equivalence relation and find the quotient set A/(R 2 R 3 ). 24

25 Solution: M R 1 1 = M R1 R 1 1 = , M R2 R 3 = = M 2, M R 1 1 R 1 =, M R 2 = Let R be a relation on Z defined by xry if x + y is an even integer. (a) Show that R is an equivalence relation on Z. (b) Find all equivalence classes of the relation R = M 3. Proof. (a) Since x + x = 2x is even for any x, then xrx, so R is reflexive. If xry, then x + y is even. Of course, y + x is even, i.e., yrx. So R is symmetric. If xry and yrz, then x + y and y + z are even, so xrz. Thus R is reflexive. Therefore R is an equivalence relation. (b) Note that xry if and only if x and y are both odd or both even. Thus there are only two equivalence classes: the set of even integers, and the set of odd integers. 7. Let X = {1, 2,..., 10} and let R be a relation on X such that arb if and only if a b 2. Determine whether R is an equivalence relation. Let M R be the matrix of R. Compute M 8 R. Solution: The following is the graph of the relation

26 Then MR 5 is a Boolean matrix all whose entries are 1. Thus M R 8 is the same as MR A relation R on a set X is called a preference relation if R is reflexive and transitive. Show that R R 1 is an equivalence relation. Proof. Obviously, R R 1 is reflexive. If x(r R 1 )y, then xry and xr 1 y, i.e., yr 1 x and yrx. Hence y(r R 1 )x. So R R 1 is symmetric. If x(r R 1 )y and y(r R 1 )z, then xry, yrz, yrx, zry, thus xrz and zrx, i.e., xrz and xr 1 z. Therefore x(r R 1 )z. So R R 1 is transitive. 9. Let n be a positive integer. The congruence relation of modulo n is an equivalence relation on Z. Let Z n denote the quotient set Z/. For any integer a Z, we define a function f a : Z n Z n, (a) Find the cardinality of the set f a (Z n ). f a ([x]) = [ax]. (b) Find all integers a such that f a is invertible. Solution: (a) Let d = gcd(a, n), a = kd, n = ld. Let x = ql + r. Then ax = kd(ql + r) = kdql + kdr = kqn + ar ar(mod n). For 1 r 1 < r 2 < l, we claim that ar 1 ar 2 (mod n). In fact, if ar 1 ar 2 (mod n), then n a(r 2 r 1 ) or equivalently l k(r 2 r 1 ). Since gcd(k, l) = 1, we have l (r 2 r 1 ). Thus r 1 = r 2, a contradiction. Therefore n f a (Z n ) = l = gcd(a, n). (b) Since Z n is finite, then f a is a bijection if and only if f a is onto. However, f a is onto if and only if f a (Z n ) = n, i.e., gcd(a, n) = For a positive integer n, let φ(n) be the number of positive integers x n such that gcd(x, n) = 1, called Euler s function. Let R be the relation on X = {1, 2,..., n} defined by xry x y, y n, gcd(x, y) = 1. 26

27 (a) Find the cardinality R 1 (y) for each y X. (b) Show that R = φ(x). x n (c) Prove R = n by showing that the function f : R X, defined by f(x, y) = xn y, is a bijection. Solution: (a) For each y X and y n, we have R 1 (y) = {x X x y, gcd(x, y) = 1} = φ(y). (b) It follows obviously that R = y X R 1 (y) = y 1, y n R 1 (y) = y n φ(y). (c) It is clear that the function f is well-defined. For x 1 Ry 1 and x 2 Ry 2, if = x 2n y 2, i.e., x 1 y 1 = x 2 y 2, then (x 1, y 1 ) and (x 2, y 2 ) are integral proportional, x 1 n y 1 say, (x 2, y 2 ) = c(x 1, y 1 ). Since gcd(x 1, y 1 ) = gcd(x 2, y 2 ) = 1, then c = 1. We thus have (x 2, y 2 ) = (x 1, y 1 ). On the other hand, for any z X, let d = gcd(z, n). Then ( z f d, n = d) (z/d) n = z. n/d This means that f is surjective. Thus f is a bijection. 11. Let X be a set of n elements. Show that the number of equivalence relations on X is n n ( 1) k (l k) n k!(l k)!. k=0 l=k (Hint: Each equivalence relation corresponds to a partition. Counting number of equivalence relations is the same as counting number of partitions.) Proof. Note that partitions of X with k parts are in one-to-one correspondent with surjective functions from X to {1, 2,..., k}. By the inclusion-exclusion 27

28 principle, the number of such surjective functions is k ( ) k ( 1) i (k i) n i k=1 i=0 Thus the answer is given by n k ( ) k k! ( 1) i (k i) n = i i=0 (Note that for k = 0, the sum k i=0 ( 1)i ( k i n ( 1) i i=0 n k=i ) (k i) n = 0.) (k i) n i!(k i)!. 28

3. Equivalence Relations. Discussion

3. Equivalence Relations. Discussion 3. EQUIVALENCE RELATIONS 33 3. Equivalence Relations 3.1. Definition of an Equivalence Relations. Definition 3.1.1. A relation R on a set A is an equivalence relation if and only if R is reflexive, symmetric,

More information

Relations Graphical View

Relations Graphical View Relations Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry Introduction Recall that a relation between elements of two sets is a subset of their Cartesian product (of ordered pairs). A binary

More information

Chapter 3. Cartesian Products and Relations. 3.1 Cartesian Products

Chapter 3. Cartesian Products and Relations. 3.1 Cartesian Products Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing

More information

Lecture 17 : Equivalence and Order Relations DRAFT

Lecture 17 : Equivalence and Order Relations DRAFT CS/Math 240: Introduction to Discrete Mathematics 3/31/2011 Lecture 17 : Equivalence and Order Relations Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT Last lecture we introduced the notion

More information

Lecture 16 : Relations and Functions DRAFT

Lecture 16 : Relations and Functions DRAFT CS/Math 240: Introduction to Discrete Mathematics 3/29/2011 Lecture 16 : Relations and Functions Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT In Lecture 3, we described a correspondence

More information

Discrete Mathematics. Hans Cuypers. October 11, 2007

Discrete Mathematics. Hans Cuypers. October 11, 2007 Hans Cuypers October 11, 2007 1 Contents 1. Relations 4 1.1. Binary relations................................ 4 1.2. Equivalence relations............................. 6 1.3. Relations and Directed Graphs.......................

More information

Chapter 10. Abstract algebra

Chapter 10. Abstract algebra Chapter 10. Abstract algebra C.O.S. Sorzano Biomedical Engineering December 17, 2013 10. Abstract algebra December 17, 2013 1 / 62 Outline 10 Abstract algebra Sets Relations and functions Partitions and

More information

Elementary Number Theory We begin with a bit of elementary number theory, which is concerned

Elementary Number Theory We begin with a bit of elementary number theory, which is concerned CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,

More information

Mathematics for Computer Science/Software Engineering. Notes for the course MSM1F3 Dr. R. A. Wilson

Mathematics for Computer Science/Software Engineering. Notes for the course MSM1F3 Dr. R. A. Wilson Mathematics for Computer Science/Software Engineering Notes for the course MSM1F3 Dr. R. A. Wilson October 1996 Chapter 1 Logic Lecture no. 1. We introduce the concept of a proposition, which is a statement

More information

I. GROUPS: BASIC DEFINITIONS AND EXAMPLES

I. GROUPS: BASIC DEFINITIONS AND EXAMPLES I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called

More information

Logic & Discrete Math in Software Engineering (CAS 701) Dr. Borzoo Bonakdarpour

Logic & Discrete Math in Software Engineering (CAS 701) Dr. Borzoo Bonakdarpour Logic & Discrete Math in Software Engineering (CAS 701) Background Dr. Borzoo Bonakdarpour Department of Computing and Software McMaster University Dr. Borzoo Bonakdarpour Logic & Discrete Math in SE (CAS

More information

Classical Analysis I

Classical Analysis I Classical Analysis I 1 Sets, relations, functions A set is considered to be a collection of objects. The objects of a set A are called elements of A. If x is an element of a set A, we write x A, and if

More information

Math 3000 Running Glossary

Math 3000 Running Glossary Math 3000 Running Glossary Last Updated on: July 15, 2014 The definition of items marked with a must be known precisely. Chapter 1: 1. A set: A collection of objects called elements. 2. The empty set (

More information

Assignment 7; Due Friday, November 11

Assignment 7; Due Friday, November 11 Assignment 7; Due Friday, November 9.8 a The set Q is not connected because we can write it as a union of two nonempty disjoint open sets, for instance U = (, 2) and V = ( 2, ). The connected subsets are

More information

SETS, RELATIONS, AND FUNCTIONS

SETS, RELATIONS, AND FUNCTIONS September 27, 2009 and notations Common Universal Subset and Power Set Cardinality Operations A set is a collection or group of objects or elements or members (Cantor 1895). the collection of the four

More information

Sets and functions. {x R : x > 0}.

Sets and functions. {x R : x > 0}. Sets and functions 1 Sets The language of sets and functions pervades mathematics, and most of the important operations in mathematics turn out to be functions or to be expressible in terms of functions.

More information

Sets, Relations and Functions

Sets, Relations and Functions Sets, Relations and Functions Eric Pacuit Department of Philosophy University of Maryland, College Park pacuit.org epacuit@umd.edu ugust 26, 2014 These notes provide a very brief background in discrete

More information

In mathematics there are endless ways that two entities can be related

In mathematics there are endless ways that two entities can be related CHAPTER 11 Relations In mathematics there are endless ways that two entities can be related to each other. Consider the following mathematical statements. 5 < 10 5 5 6 = 30 5 5 80 7 > 4 x y 8 3 a b ( mod

More information

Diagonal, Symmetric and Triangular Matrices

Diagonal, Symmetric and Triangular Matrices Contents 1 Diagonal, Symmetric Triangular Matrices 2 Diagonal Matrices 2.1 Products, Powers Inverses of Diagonal Matrices 2.1.1 Theorem (Powers of Matrices) 2.2 Multiplying Matrices on the Left Right by

More information

MAT2400 Analysis I. A brief introduction to proofs, sets, and functions

MAT2400 Analysis I. A brief introduction to proofs, sets, and functions MAT2400 Analysis I A brief introduction to proofs, sets, and functions In Analysis I there is a lot of manipulations with sets and functions. It is probably also the first course where you have to take

More information

In mathematics you don t understand things. You just get used to them. (Attributed to John von Neumann)

In mathematics you don t understand things. You just get used to them. (Attributed to John von Neumann) Chapter 1 Sets and Functions We understand a set to be any collection M of certain distinct objects of our thought or intuition (called the elements of M) into a whole. (Georg Cantor, 1895) In mathematics

More information

This chapter describes set theory, a mathematical theory that underlies all of modern mathematics.

This chapter describes set theory, a mathematical theory that underlies all of modern mathematics. Appendix A Set Theory This chapter describes set theory, a mathematical theory that underlies all of modern mathematics. A.1 Basic Definitions Definition A.1.1. A set is an unordered collection of elements.

More information

Further linear algebra. Chapter I. Integers.

Further linear algebra. Chapter I. Integers. Further linear algebra. Chapter I. Integers. Andrei Yafaev Number theory is the theory of Z = {0, ±1, ±2,...}. 1 Euclid s algorithm, Bézout s identity and the greatest common divisor. We say that a Z divides

More information

POWER SETS AND RELATIONS

POWER SETS AND RELATIONS POWER SETS AND RELATIONS L. MARIZZA A. BAILEY 1. The Power Set Now that we have defined sets as best we can, we can consider a sets of sets. If we were to assume nothing, except the existence of the empty

More information

CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE

CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE With the notion of bijection at hand, it is easy to formalize the idea that two finite sets have the same number of elements: we just need to verify

More information

vertex, 369 disjoint pairwise, 395 disjoint sets, 236 disjunction, 33, 36 distributive laws

vertex, 369 disjoint pairwise, 395 disjoint sets, 236 disjunction, 33, 36 distributive laws Index absolute value, 135 141 additive identity, 254 additive inverse, 254 aleph, 466 algebra of sets, 245, 278 antisymmetric relation, 387 arcsine function, 349 arithmetic sequence, 208 arrow diagram,

More information

S(A) X α for all α Λ. Consequently, S(A) X, by the definition of intersection. Therefore, X is inductive.

S(A) X α for all α Λ. Consequently, S(A) X, by the definition of intersection. Therefore, X is inductive. MA 274: Exam 2 Study Guide (1) Know the precise definitions of the terms requested for your journal. (2) Review proofs by induction. (3) Be able to prove that something is or isn t an equivalence relation.

More information

INTRODUCTORY SET THEORY

INTRODUCTORY SET THEORY M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H-1088 Budapest, Múzeum krt. 6-8. CONTENTS 1. SETS Set, equal sets, subset,

More information

MATH 289 PROBLEM SET 1: INDUCTION. 1. The induction Principle The following property of the natural numbers is intuitively clear:

MATH 289 PROBLEM SET 1: INDUCTION. 1. The induction Principle The following property of the natural numbers is intuitively clear: MATH 89 PROBLEM SET : INDUCTION The induction Principle The following property of the natural numbers is intuitively clear: Axiom Every nonempty subset of the set of nonnegative integers Z 0 = {0,,, 3,

More information

7 Relations and Partial Orders

7 Relations and Partial Orders mcs-ftl 2010/9/8 0:40 page 213 #219 7 Relations and Partial Orders A relation is a mathematical tool for describing associations between elements of sets. Relations are widely used in computer science,

More information

Sets and Cardinality Notes for C. F. Miller

Sets and Cardinality Notes for C. F. Miller Sets and Cardinality Notes for 620-111 C. F. Miller Semester 1, 2000 Abstract These lecture notes were compiled in the Department of Mathematics and Statistics in the University of Melbourne for the use

More information

p 2 1 (mod 6) Adding 2 to both sides gives p (mod 6)

p 2 1 (mod 6) Adding 2 to both sides gives p (mod 6) .9. Problems P10 Try small prime numbers first. p p + 6 3 11 5 7 7 51 11 13 Among the primes in this table, only the prime 3 has the property that (p + ) is also a prime. We try to prove that no other

More information

Problem Set 1 Solutions Math 109

Problem Set 1 Solutions Math 109 Problem Set 1 Solutions Math 109 Exercise 1.6 Show that a regular tetrahedron has a total of twenty-four symmetries if reflections and products of reflections are allowed. Identify a symmetry which is

More information

MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set.

MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. Vector space A vector space is a set V equipped with two operations, addition V V (x,y) x + y V and scalar

More information

Quotient Rings and Field Extensions

Quotient Rings and Field Extensions Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.

More information

Cartesian Products and Relations

Cartesian Products and Relations Cartesian Products and Relations Definition (Cartesian product) If A and B are sets, the Cartesian product of A and B is the set A B = {(a, b) :(a A) and (b B)}. The following points are worth special

More information

Notes on Matrix Multiplication and the Transitive Closure

Notes on Matrix Multiplication and the Transitive Closure ICS 6D Due: Wednesday, February 25, 2015 Instructor: Sandy Irani Notes on Matrix Multiplication and the Transitive Closure An n m matrix over a set S is an array of elements from S with n rows and m columns.

More information

Equivalence Relations

Equivalence Relations Equivalence Relations Definition An equivalence relation on a set S, is a relation on S which is reflexive, symmetric and transitive. Examples: Let S = Z and define R = {(x,y) x and y have the same parity}

More information

8 Divisibility and prime numbers

8 Divisibility and prime numbers 8 Divisibility and prime numbers 8.1 Divisibility In this short section we extend the concept of a multiple from the natural numbers to the integers. We also summarize several other terms that express

More information

10. Graph Matrices Incidence Matrix

10. Graph Matrices Incidence Matrix 10 Graph Matrices Since a graph is completely determined by specifying either its adjacency structure or its incidence structure, these specifications provide far more efficient ways of representing a

More information

Homework MA 725 Spring, 2012 C. Huneke SELECTED ANSWERS

Homework MA 725 Spring, 2012 C. Huneke SELECTED ANSWERS Homework MA 725 Spring, 2012 C. Huneke SELECTED ANSWERS 1.1.25 Prove that the Petersen graph has no cycle of length 7. Solution: There are 10 vertices in the Petersen graph G. Assume there is a cycle C

More information

Long questions answer Advanced Mathematics for Computer Application If P= , find BT. 19. If B = 1 0, find 2B and -3B.

Long questions answer Advanced Mathematics for Computer Application If P= , find BT. 19. If B = 1 0, find 2B and -3B. Unit-1: Matrix Algebra Short questions answer 1. What is Matrix? 2. Define the following terms : a) Elements matrix b) Row matrix c) Column matrix d) Diagonal matrix e) Scalar matrix f) Unit matrix OR

More information

Practice Problems for First Test

Practice Problems for First Test Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate.-

More information

Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

More information

Introduction to Flocking {Stochastic Matrices}

Introduction to Flocking {Stochastic Matrices} Supelec EECI Graduate School in Control Introduction to Flocking {Stochastic Matrices} A. S. Morse Yale University Gif sur - Yvette May 21, 2012 CRAIG REYNOLDS - 1987 BOIDS The Lion King CRAIG REYNOLDS

More information

= 2 + 1 2 2 = 3 4, Now assume that P (k) is true for some fixed k 2. This means that

= 2 + 1 2 2 = 3 4, Now assume that P (k) is true for some fixed k 2. This means that Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without

More information

CHAPTER 5. Number Theory. 1. Integers and Division. Discussion

CHAPTER 5. Number Theory. 1. Integers and Division. Discussion CHAPTER 5 Number Theory 1. Integers and Division 1.1. Divisibility. Definition 1.1.1. Given two integers a and b we say a divides b if there is an integer c such that b = ac. If a divides b, we write a

More information

NOTES ON LINEAR TRANSFORMATIONS

NOTES ON LINEAR TRANSFORMATIONS NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all

More information

A matrix over a field F is a rectangular array of elements from F. The symbol

A matrix over a field F is a rectangular array of elements from F. The symbol Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F) denotes the collection of all m n matrices over F Matrices will usually be denoted

More information

MODULAR ARITHMETIC. a smallest member. It is equivalent to the Principle of Mathematical Induction.

MODULAR ARITHMETIC. a smallest member. It is equivalent to the Principle of Mathematical Induction. MODULAR ARITHMETIC 1 Working With Integers The usual arithmetic operations of addition, subtraction and multiplication can be performed on integers, and the result is always another integer Division, on

More information

6. Metric spaces. In this section we review the basic facts about metric spaces. d : X X [0, )

6. Metric spaces. In this section we review the basic facts about metric spaces. d : X X [0, ) 6. Metric spaces In this section we review the basic facts about metric spaces. Definitions. A metric on a non-empty set X is a map with the following properties: d : X X [0, ) (i) If x, y X are points

More information

Course 421: Algebraic Topology Section 1: Topological Spaces

Course 421: Algebraic Topology Section 1: Topological Spaces Course 421: Algebraic Topology Section 1: Topological Spaces David R. Wilkins Copyright c David R. Wilkins 1988 2008 Contents 1 Topological Spaces 1 1.1 Continuity and Topological Spaces...............

More information

Homework until Test #2

Homework until Test #2 MATH31: Number Theory Homework until Test # Philipp BRAUN Section 3.1 page 43, 1. It has been conjectured that there are infinitely many primes of the form n. Exhibit five such primes. Solution. Five such

More information

GROUPS SUBGROUPS. Definition 1: An operation on a set G is a function : G G G.

GROUPS SUBGROUPS. Definition 1: An operation on a set G is a function : G G G. Definition 1: GROUPS An operation on a set G is a function : G G G. Definition 2: A group is a set G which is equipped with an operation and a special element e G, called the identity, such that (i) the

More information

a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.

a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2. Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given

More information

THE NUMBER OF GRAPHS AND A RANDOM GRAPH WITH A GIVEN DEGREE SEQUENCE. Alexander Barvinok

THE NUMBER OF GRAPHS AND A RANDOM GRAPH WITH A GIVEN DEGREE SEQUENCE. Alexander Barvinok THE NUMBER OF GRAPHS AND A RANDOM GRAPH WITH A GIVEN DEGREE SEQUENCE Alexer Barvinok Papers are available at http://www.math.lsa.umich.edu/ barvinok/papers.html This is a joint work with J.A. Hartigan

More information

Notes on Algebraic Structures. Peter J. Cameron

Notes on Algebraic Structures. Peter J. Cameron Notes on Algebraic Structures Peter J. Cameron ii Preface These are the notes of the second-year course Algebraic Structures I at Queen Mary, University of London, as I taught it in the second semester

More information

1.1 Logical Form and Logical Equivalence 1

1.1 Logical Form and Logical Equivalence 1 Contents Chapter I The Logic of Compound Statements 1.1 Logical Form and Logical Equivalence 1 Identifying logical form; Statements; Logical connectives: not, and, and or; Translation to and from symbolic

More information

1. LINEAR EQUATIONS. A linear equation in n unknowns x 1, x 2,, x n is an equation of the form

1. LINEAR EQUATIONS. A linear equation in n unknowns x 1, x 2,, x n is an equation of the form 1. LINEAR EQUATIONS A linear equation in n unknowns x 1, x 2,, x n is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b, where a 1, a 2,..., a n, b are given real numbers. For example, with x and

More information

Partitioning edge-coloured complete graphs into monochromatic cycles and paths

Partitioning edge-coloured complete graphs into monochromatic cycles and paths arxiv:1205.5492v1 [math.co] 24 May 2012 Partitioning edge-coloured complete graphs into monochromatic cycles and paths Alexey Pokrovskiy Departement of Mathematics, London School of Economics and Political

More information

ORDERS OF ELEMENTS IN A GROUP

ORDERS OF ELEMENTS IN A GROUP ORDERS OF ELEMENTS IN A GROUP KEITH CONRAD 1. Introduction Let G be a group and g G. We say g has finite order if g n = e for some positive integer n. For example, 1 and i have finite order in C, since

More information

Lemma 5.2. Let S be a set. (1) Let f and g be two permutations of S. Then the composition of f and g is a permutation of S.

Lemma 5.2. Let S be a set. (1) Let f and g be two permutations of S. Then the composition of f and g is a permutation of S. Definition 51 Let S be a set bijection f : S S 5 Permutation groups A permutation of S is simply a Lemma 52 Let S be a set (1) Let f and g be two permutations of S Then the composition of f and g is a

More information

Matrix Representations of Linear Transformations and Changes of Coordinates

Matrix Representations of Linear Transformations and Changes of Coordinates Matrix Representations of Linear Transformations and Changes of Coordinates 01 Subspaces and Bases 011 Definitions A subspace V of R n is a subset of R n that contains the zero element and is closed under

More information

1. Determine all real numbers a, b, c, d that satisfy the following system of equations.

1. Determine all real numbers a, b, c, d that satisfy the following system of equations. altic Way 1999 Reykjavík, November 6, 1999 Problems 1. etermine all real numbers a, b, c, d that satisfy the following system of equations. abc + ab + bc + ca + a + b + c = 1 bcd + bc + cd + db + b + c

More information

CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY

CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY January 10, 2010 CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY The set of polynomials over a field F is a ring, whose structure shares with the ring of integers many characteristics.

More information

Discrete Maths. Philippa Gardner. These lecture notes are based on previous notes by Iain Phillips.

Discrete Maths. Philippa Gardner. These lecture notes are based on previous notes by Iain Phillips. Discrete Maths Philippa Gardner These lecture notes are based on previous notes by Iain Phillips. This short course introduces some basic concepts in discrete mathematics: sets, relations, and functions.

More information

Systems of Linear Equations

Systems of Linear Equations Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and

More information

1. Prove that the empty set is a subset of every set.

1. Prove that the empty set is a subset of every set. 1. Prove that the empty set is a subset of every set. Basic Topology Written by Men-Gen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since

More information

CHAPTER 5: MODULAR ARITHMETIC

CHAPTER 5: MODULAR ARITHMETIC CHAPTER 5: MODULAR ARITHMETIC LECTURE NOTES FOR MATH 378 (CSUSM, SPRING 2009). WAYNE AITKEN 1. Introduction In this chapter we will consider congruence modulo m, and explore the associated arithmetic called

More information

R u t c o r Research R e p o r t. Boolean functions with a simple certificate for CNF complexity. Ondřej Čepeka Petr Kučera b Petr Savický c

R u t c o r Research R e p o r t. Boolean functions with a simple certificate for CNF complexity. Ondřej Čepeka Petr Kučera b Petr Savický c R u t c o r Research R e p o r t Boolean functions with a simple certificate for CNF complexity Ondřej Čepeka Petr Kučera b Petr Savický c RRR 2-2010, January 24, 2010 RUTCOR Rutgers Center for Operations

More information

Discrete Mathematics, Chapter 5: Induction and Recursion

Discrete Mathematics, Chapter 5: Induction and Recursion Discrete Mathematics, Chapter 5: Induction and Recursion Richard Mayr University of Edinburgh, UK Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 5 1 / 20 Outline 1 Well-founded

More information

Matrix Algebra 2.3 CHARACTERIZATIONS OF INVERTIBLE MATRICES Pearson Education, Inc.

Matrix Algebra 2.3 CHARACTERIZATIONS OF INVERTIBLE MATRICES Pearson Education, Inc. 2 Matrix Algebra 2.3 CHARACTERIZATIONS OF INVERTIBLE MATRICES Theorem 8: Let A be a square matrix. Then the following statements are equivalent. That is, for a given A, the statements are either all true

More information

Introducing Functions

Introducing Functions Functions 1 Introducing Functions A function f from a set A to a set B, written f : A B, is a relation f A B such that every element of A is related to one element of B; in logical notation 1. (a, b 1

More information

26 Ideals and Quotient Rings

26 Ideals and Quotient Rings Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 26 Ideals and Quotient Rings In this section we develop some theory of rings that parallels the theory of groups discussed

More information

5.1 Commutative rings; Integral Domains

5.1 Commutative rings; Integral Domains 5.1 J.A.Beachy 1 5.1 Commutative rings; Integral Domains from A Study Guide for Beginner s by J.A.Beachy, a supplement to Abstract Algebra by Beachy / Blair 23. Let R be a commutative ring. Prove the following

More information

CS 103X: Discrete Structures Homework Assignment 3 Solutions

CS 103X: Discrete Structures Homework Assignment 3 Solutions CS 103X: Discrete Structures Homework Assignment 3 s Exercise 1 (20 points). On well-ordering and induction: (a) Prove the induction principle from the well-ordering principle. (b) Prove the well-ordering

More information

Preference Relations and Choice Rules

Preference Relations and Choice Rules Preference Relations and Choice Rules Econ 2100, Fall 2015 Lecture 1, 31 August Outline 1 Logistics 2 Binary Relations 1 Definition 2 Properties 3 Upper and Lower Contour Sets 3 Preferences 4 Choice Correspondences

More information

Lecture 13 - Basic Number Theory.

Lecture 13 - Basic Number Theory. Lecture 13 - Basic Number Theory. Boaz Barak March 22, 2010 Divisibility and primes Unless mentioned otherwise throughout this lecture all numbers are non-negative integers. We say that A divides B, denoted

More information

CONTENTS 1. Peter Kahn. Spring 2007

CONTENTS 1. Peter Kahn. Spring 2007 CONTENTS 1 MATH 304: CONSTRUCTING THE REAL NUMBERS Peter Kahn Spring 2007 Contents 2 The Integers 1 2.1 The basic construction.......................... 1 2.2 Adding integers..............................

More information

Review for Final Exam

Review for Final Exam Review for Final Exam Note: Warning, this is probably not exhaustive and probably does contain typos (which I d like to hear about), but represents a review of most of the material covered in Chapters

More information

Revision of ring theory

Revision of ring theory CHAPTER 1 Revision of ring theory 1.1. Basic definitions and examples In this chapter we will revise and extend some of the results on rings that you have studied on previous courses. A ring is an algebraic

More information

Degree Hypergroupoids Associated with Hypergraphs

Degree Hypergroupoids Associated with Hypergraphs Filomat 8:1 (014), 119 19 DOI 10.98/FIL1401119F Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Degree Hypergroupoids Associated

More information

. 0 1 10 2 100 11 1000 3 20 1 2 3 4 5 6 7 8 9

. 0 1 10 2 100 11 1000 3 20 1 2 3 4 5 6 7 8 9 Introduction The purpose of this note is to find and study a method for determining and counting all the positive integer divisors of a positive integer Let N be a given positive integer We say d is a

More information

Solutions to In-Class Problems Week 4, Mon.

Solutions to In-Class Problems Week 4, Mon. Massachusetts Institute of Technology 6.042J/18.062J, Fall 05: Mathematics for Computer Science September 26 Prof. Albert R. Meyer and Prof. Ronitt Rubinfeld revised September 26, 2005, 1050 minutes Solutions

More information

UNIT 2 MATRICES - I 2.0 INTRODUCTION. Structure

UNIT 2 MATRICES - I 2.0 INTRODUCTION. Structure UNIT 2 MATRICES - I Matrices - I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress

More information

Pythagorean Triples, Complex Numbers, Abelian Groups and Prime Numbers

Pythagorean Triples, Complex Numbers, Abelian Groups and Prime Numbers Pythagorean Triples, Complex Numbers, Abelian Groups and Prime Numbers Amnon Yekutieli Department of Mathematics Ben Gurion University email: amyekut@math.bgu.ac.il Notes available at http://www.math.bgu.ac.il/~amyekut/lectures

More information

Zachary Monaco Georgia College Olympic Coloring: Go For The Gold

Zachary Monaco Georgia College Olympic Coloring: Go For The Gold Zachary Monaco Georgia College Olympic Coloring: Go For The Gold Coloring the vertices or edges of a graph leads to a variety of interesting applications in graph theory These applications include various

More information

Chapter 7: Products and quotients

Chapter 7: Products and quotients Chapter 7: Products and quotients Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 42, Spring 24 M. Macauley (Clemson) Chapter 7: Products

More information

Discrete Mathematics & Mathematical Reasoning Chapter 10: Graphs

Discrete Mathematics & Mathematical Reasoning Chapter 10: Graphs Discrete Mathematics & Mathematical Reasoning Chapter 10: Graphs Kousha Etessami U. of Edinburgh, UK Kousha Etessami (U. of Edinburgh, UK) Discrete Mathematics (Chapter 6) 1 / 13 Overview Graphs and Graph

More information

1 Limiting distribution for a Markov chain

1 Limiting distribution for a Markov chain Copyright c 2009 by Karl Sigman Limiting distribution for a Markov chain In these Lecture Notes, we shall study the limiting behavior of Markov chains as time n In particular, under suitable easy-to-check

More information

SUM OF TWO SQUARES JAHNAVI BHASKAR

SUM OF TWO SQUARES JAHNAVI BHASKAR SUM OF TWO SQUARES JAHNAVI BHASKAR Abstract. I will investigate which numbers can be written as the sum of two squares and in how many ways, providing enough basic number theory so even the unacquainted

More information

PROBLEM SET 7: PIGEON HOLE PRINCIPLE

PROBLEM SET 7: PIGEON HOLE PRINCIPLE PROBLEM SET 7: PIGEON HOLE PRINCIPLE The pigeonhole principle is the following observation: Theorem. Suppose that > kn marbles are distributed over n jars, then one jar will contain at least k + marbles.

More information

ABSTRACT ALGEBRA: A STUDY GUIDE FOR BEGINNERS

ABSTRACT ALGEBRA: A STUDY GUIDE FOR BEGINNERS ABSTRACT ALGEBRA: A STUDY GUIDE FOR BEGINNERS John A. Beachy Northern Illinois University 2014 ii J.A.Beachy This is a supplement to Abstract Algebra, Third Edition by John A. Beachy and William D. Blair

More information

Algorithmic Mathematics

Algorithmic Mathematics Algorithmic Mathematics a web-book by Leonard Soicher & Franco Vivaldi This is the textbook for the course MAS202 Algorithmic Mathematics. This material is in a fluid state it is rapidly evolving and as

More information

SCORE SETS IN ORIENTED GRAPHS

SCORE SETS IN ORIENTED GRAPHS Applicable Analysis and Discrete Mathematics, 2 (2008), 107 113. Available electronically at http://pefmath.etf.bg.ac.yu SCORE SETS IN ORIENTED GRAPHS S. Pirzada, T. A. Naikoo The score of a vertex v in

More information

Abstract Algebra Theory and Applications. Thomas W. Judson Stephen F. Austin State University

Abstract Algebra Theory and Applications. Thomas W. Judson Stephen F. Austin State University Abstract Algebra Theory and Applications Thomas W. Judson Stephen F. Austin State University August 16, 2013 ii Copyright 1997-2013 by Thomas W. Judson. Permission is granted to copy, distribute and/or

More information

Every Positive Integer is the Sum of Four Squares! (and other exciting problems)

Every Positive Integer is the Sum of Four Squares! (and other exciting problems) Every Positive Integer is the Sum of Four Squares! (and other exciting problems) Sophex University of Texas at Austin October 18th, 00 Matilde N. Lalín 1. Lagrange s Theorem Theorem 1 Every positive integer

More information

MATHEMATICAL INDUCTION. Mathematical Induction. This is a powerful method to prove properties of positive integers.

MATHEMATICAL INDUCTION. Mathematical Induction. This is a powerful method to prove properties of positive integers. MATHEMATICAL INDUCTION MIGUEL A LERMA (Last updated: February 8, 003) Mathematical Induction This is a powerful method to prove properties of positive integers Principle of Mathematical Induction Let P

More information

Introduction to Topology

Introduction to Topology Introduction to Topology Tomoo Matsumura November 30, 2010 Contents 1 Topological spaces 3 1.1 Basis of a Topology......................................... 3 1.2 Comparing Topologies.......................................

More information

Breaking The Code. Ryan Lowe. Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and

Breaking The Code. Ryan Lowe. Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and Breaking The Code Ryan Lowe Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and a minor in Applied Physics. As a sophomore, he took an independent study

More information