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Problem 3. Which of the following relations are equivalence relations on the given set S (1)...
2. Consider the set A = {1, 2, 3, ... , 12} and the relations Em...
2. Consider the set A = {1, 2, 3, ... , 12} and the relations Em on A where x =m y means m divides x – y. (These are equivalence relations on A for the same reason as the similarly-defined relations on all of Z.) For each x E A, find the equivalence classes [x]=ş and [x]=4. Which =3 -equivalence classes are the same? Which 34 -equivalence classes are the same?
probelms 9.1 9 Modular arithmetic Definition 9.1 Let S be a set. A relation R =...
probelms 9.1
9 Modular arithmetic Definition 9.1 Let S be a set. A relation R = R(,y) on S is a statement about pairs (x,y) of elements of S. For r,y ES, I is related to y notation: Ry) if R(x,y) is true. A relation Ris: Reflexive if for any I ES, R. Symmetric if for any ry ES, Ry implies y Rr. Transitive if for any r.y.ES, Ry and yRimply R. An equivalence relation is a reflexive, symmetric and...
4. Find the equivalence classes for each of the following equivalence relations R on the given...
4. Find the equivalence classes for each of the following equivalence relations R on the given sets A: (i) R = {(a,b): a = b (modulo 6)}, where A is the set of non-negative integers; (ii) R = {(0,0), (2,2), (0,2), (2,0), (4,4), (6,6), (6,8), (6,9), (8,6), (8,8), (8,9), (9,6), (9,8), (9,9)}, where A = {0, 2, 4, 6, 8, 9}
1. (14 points) (a) For each of the following relations R on the given domains A,...
1. (14 points) (a) For each of the following relations R on the given domains A, categorize them as not an equivalence relation, an equivalence relation with finitely many distinct equivalence classes, or an equivalence relation with infinitely many distinct equivalence classes. Justify each decision with a brief proof. (i) A = {1, 2, 3} , R = {(1, 1),(2, 2),(3, 3)} (ii) A = R, R = {(x, y) | x 2 = y 2} (iii) A = Z,...
(1) Suppose R and S are reflexive relations on a set A. Prove or disprove each...
(1) Suppose R and S are reflexive relations on a set A. Prove or disprove each of these statements. (a) RUS is reflexive. (b) Rn S is reflexive. (c) R\S is reflexive. (2) Define the equivalence relation on the set Z where a ~b if and only if a? = 62. (a) List the element(s) of 7. (b) List the element(s) of -1. (c) Describe the set of all equivalence classes.
10. Verify that the relations given below are quasiorders. List the elements of each equivalence class of the induced equivalence relation, and draw the Hasse (a) On the set (1,2,..., 303, define...
10. Verify that the relations given below are quasiorders. List the elements of each equivalence class of the induced equivalence relation, and draw the Hasse (a) On the set (1,2,..., 303, define mn if and only if the sum of the digits (b) On the set (1.2,3,4,11, 12, 13,14,21,22,23,24), define mn if and only diagram for the induced partial order on the equivalence classes of m is less than or equal to the sum of the digits of n. if...
QUESTION 10 The equality relationon any set S is: A total ordering and a function with an inverse. An equivalence relation and also function with an inverse. A function with an inverse, and an equiva...
QUESTION 10 The equality relationon any set S is: A total ordering and a function with an inverse. An equivalence relation and also function with an inverse. A function with an inverse, and an equivalence relation with as single equivalence class equal to S An equivalence relation and also a total ordering QUESTION 11 A binary operation on a set S, takes any two elements a,b E S and produces another element c e S. Examples of binary operations include...
Please solve the 3 questions. Question 1: Determine the set of the points at which each...
Please solve the 3 questions.
Question 1: Determine the set of the points at which each given function is continuous: 1) f(x,y) = (5 Marks 4-x2 - y2 (2) g(x,y,z) = In x2 + y2 - 61 Cosz (x2 - y2 + z) 3) h(x,y,z) = x + 2y - 2 x2 + y2 1
Math 240 Assignment 4 - due Friday, February 28 each relation R defined on the given...
Math 240 Assignment 4 - due Friday, February 28 each relation R defined on the given set A, determine whether or not it is reflexive, symmetric, anti-symmetric, or transitive. Explain why. (a) A = {0, 1,2,3), R = {(0,0).(0,1),(1,1),(1,2).(2, 2), (2.3)} (b) A = {0, 1,2,3), R = {(0,0).(0,2), (1,1),(1,3), (2,0), (2,2), (3,1),(3,3)} (c) A is the set of all English words. For words a and b, (a,b) E R if and only if a and b have at least...
8. Let S = classes? 1, 2, 3, 4, 5, 6, 7, 8). How many equivalence relations on S have exactly 3 e...
8. Let S = classes? 1, 2, 3, 4, 5, 6, 7, 8). How many equivalence relations on S have exactly 3 equivalence
8. Let S = classes? 1, 2, 3, 4, 5, 6, 7, 8). How many equivalence relations on S have exactly 3 equivalence
2. Consider the set A = {1, 2, 3, ... , 12} and the relations Em on A where x =m y means m divides x – y. (These are equivalence relations on A for the same reason as the similarly-defined relations on all of Z.) For each x E A, find the equivalence classes [x]=ş and [x]=4. Which =3 -equivalence classes are the same? Which 34 -equivalence classes are the same?
probelms 9.1
9 Modular arithmetic Definition 9.1 Let S be a set. A relation R = R(,y) on S is a statement about pairs (x,y) of elements of S. For r,y ES, I is related to y notation: Ry) if R(x,y) is true. A relation Ris: Reflexive if for any I ES, R. Symmetric if for any ry ES, Ry implies y Rr. Transitive if for any r.y.ES, Ry and yRimply R. An equivalence relation is a reflexive, symmetric and...
4. Find the equivalence classes for each of the following equivalence relations R on the given sets A: (i) R = {(a,b): a = b (modulo 6)}, where A is the set of non-negative integers; (ii) R = {(0,0), (2,2), (0,2), (2,0), (4,4), (6,6), (6,8), (6,9), (8,6), (8,8), (8,9), (9,6), (9,8), (9,9)}, where A = {0, 2, 4, 6, 8, 9}
(1) Suppose R and S are reflexive relations on a set A. Prove or disprove each of these statements. (a) RUS is reflexive. (b) Rn S is reflexive. (c) R\S is reflexive. (2) Define the equivalence relation on the set Z where a ~b if and only if a? = 62. (a) List the element(s) of 7. (b) List the element(s) of -1. (c) Describe the set of all equivalence classes.
10. Verify that the relations given below are quasiorders. List the elements of each equivalence class of the induced equivalence relation, and draw the Hasse (a) On the set (1,2,..., 303, define mn if and only if the sum of the digits (b) On the set (1.2,3,4,11, 12, 13,14,21,22,23,24), define mn if and only diagram for the induced partial order on the equivalence classes of m is less than or equal to the sum of the digits of n. if...
QUESTION 10 The equality relationon any set S is: A total ordering and a function with an inverse. An equivalence relation and also function with an inverse. A function with an inverse, and an equivalence relation with as single equivalence class equal to S An equivalence relation and also a total ordering QUESTION 11 A binary operation on a set S, takes any two elements a,b E S and produces another element c e S. Examples of binary operations include...
Please solve the 3 questions.
Question 1: Determine the set of the points at which each given function is continuous: 1) f(x,y) = (5 Marks 4-x2 - y2 (2) g(x,y,z) = In x2 + y2 - 61 Cosz (x2 - y2 + z) 3) h(x,y,z) = x + 2y - 2 x2 + y2 1
Math 240 Assignment 4 - due Friday, February 28 each relation R defined on the given set A, determine whether or not it is reflexive, symmetric, anti-symmetric, or transitive. Explain why. (a) A = {0, 1,2,3), R = {(0,0).(0,1),(1,1),(1,2).(2, 2), (2.3)} (b) A = {0, 1,2,3), R = {(0,0).(0,2), (1,1),(1,3), (2,0), (2,2), (3,1),(3,3)} (c) A is the set of all English words. For words a and b, (a,b) E R if and only if a and b have at least...
8. Let S = classes? 1, 2, 3, 4, 5, 6, 7, 8). How many equivalence relations on S have exactly 3 equivalence
8. Let S = classes? 1, 2, 3, 4, 5, 6, 7, 8). How many equivalence relations on S have exactly 3 equivalence
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