3. (a) If aRb is a relation of congruent modulo n, a ≡ b (mod n). Show that R is:
(i) reflexive.
(ii) symmetric.
(iii) transitive.
(b) A is a set and | A | = 8. R is a relation on A, R ⊆ A X A.
(i) How many different R can be produced?
(ii) How many R are reflexive?
(iii) How many R are symmetric?
(iv) How many R are reflexive and symmetric?
Homework Answers
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3. (a) If aRb is a relation of congruent modulo n, a ≡ b (mod n)....
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1. (2 marks) Let S 2,3,4,5,6,7,8,9, 10, 11, 12). Let r be the relation on the set S defined as follows: Va,bE S, arb if and only if every prime number that divides a is a factor of b and a S b. The relation T is a partial order relation (you do not need to prove this). Draw the Hasse diagram for T
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2. Let S 11,2,3,4,5, 6, 7,8,91 and let T 12,4,6,8. Let R be the relation on P (S) detined by for ...
2. Let S 11,2,3,4,5, 6, 7,8,91 and let T 12,4,6,8. Let R be the relation on P (S) detined by for all X, Y E P (s), (X, Y) E R if and only if IX-T] = IY-T]. (a) Prove that R is an equivalence relation. (b) How many equivalence classes are there? Explain. (c) How mauy elements of [ø], the equivalence class of ø, are there? Explain (d) How many elements of [f1,2,3, 4)], the equivalence class of (1,2,3,...
1) Let R be the relation defined on N N as follows: (m, n)R(p, q) if...
1) Let R be the relation defined on N N as follows: (m, n)R(p, q) if and only if m - pis divisible by 3 and n - q is divisible by 5. For example, (2, 19)R(8,4). 1. Identify two elements of N X N which are related under R to (6, 45). II. Is R reflexive? Justify your answer. III. Is R symmetric? Justify your answer. IV. Is R transitive? Justify your answer. V.Is R an equivalence relation? Justify...
Q-4. [8+3+3+3+3 marks] Let be the partial order relation defined on , where means. a) Draw...
Q-4. [8+3+3+3+3 marks] Let be the partial order relation defined on , where means. a) Draw the Hasse diagram for . b) Find all maximal and minimal elements. c) Find lub({6,12}). a) Find glb({6,12}). e) What is the least element? The greatest element? Q-4. [8+3+3+3+3 marks] Let R be the partial order relation defined on A = {2,3, 6, 9, 10, 12, 14, 18, 20}, where xRy means x|y. a) Draw the Hasse diagram for R. b) Find all maximal...
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(b) The delay time for a packet travelling between four routers (a, b, c, d) is given in the following table: To b From 1 msec 2 msec a b 2 msec 5 msec 2 msec C 5 msec 5 msec 2msec Draw the relation digraph from the information given by the above table (i) (10/100) (ii) Relation R on set {a, b, c, d} is defined as follows: "xRy if the delay time from router x to router y...
Problem 4. Let A, B e Rmxn. We say that A is equivalent to B if there exist an invertible m x m n x n matrix Q such tha...
Problem 4. Let A, B e Rmxn. We say that A is equivalent to B if there exist an invertible m x m n x n matrix Q such that PAQ = B. matrix P and an invertible (a) Prove that the relation "A is equivalent to B" is reflexive, symmetric, and transitive; i.e., prove that: (i) for all A E Rmx", A is equivalent to A; (ii) for all A, B e Rmxn, if A is equivalent to B...
I. Let each of R, S, and T be binary relations on N2 as defined here: R-[<m, n EN nis the smallest prime number greater than or equal to m] S -[< m, n> EN* nis the greatest prime number less...
I. Let each of R, S, and T be binary relations on N2 as defined here: R-[<m, n EN nis the smallest prime number greater than or equal to m] S -[< m, n> EN* nis the greatest prime number less than or equal to m] (a) Which (if any) of these binary relations is a (unary) function? (b) Which (if any) of these binary relations is an injection? (c) Which (if any) of these binary relations is a surjection?...
In java please Program 5.1: Properties of a Relation Write a program that can determine which,...
In java please
Program 5.1: Properties of a Relation Write a program that can determine which, if any, of the following properties a binary matrix exhibits: symmetric anti-symmetric asymmetric (from the textbook) reflexive or anti-reflexive (or neither of course) Program Requirements: Hard code at least 4 binary matrices all of size 4x4. Display each binary matrix and the the properties it exhibits. For example: A 0 100 0 00 0 0000 0 000 A anti-reflexive, anti-symmetric, asymmetric B 11 10...
discrete math Need 7c 9ab 10 15 16 17 (7) Consider the following matrices. Compute the...
discrete math
Need 7c 9ab 10 15 16 17
(7) Consider the following matrices. Compute the following matrices A=[ ]B=[ 1 c-[! (a) CA (b) BAA (c) AOC (9) Determine if the following statements are True or False. If the statement is False, explain why. (a) Consider A={1,2,3,4,5). Do A1 = {1,3,5}, A2 = {2,4}. (i) Show that P ={A1, A2} forms a partition of A. (ii) Construct the matrix of the relation R corresponding to P (b) Consider A...
(b) Answer the following questions for the relation R defined on the set of seven-bit strings...
(b) Answer the following questions for the relation R defined on the set of seven-bit strings by s,RS2, provided that the first four bits of s, and s2 coincide. Jawab soalan-soalan berikut untuk hubungan R yang ditakrif di set rentetan tujuh-bit oleh s1Rs2, dengan syarat bahawa empat bit pertama s1 dan s2 serentak. (i) Show that R is an equivalence relation. Tunjukkan R adalah hubungan kesetaraan. (20/100) (ii) How many equivalence classes are there? Berapa banyak kelas kesetaraan yang ada?...
1. (2 marks) Let S 2,3,4,5,6,7,8,9, 10, 11, 12). Let r be the relation on the set S defined as follows: Va,bE S, arb if and only if every prime number that divides a is a factor of b and a S b. The relation T is a partial order relation (you do not need to prove this). Draw the Hasse diagram for T
1. (2 marks) Let S 2,3,4,5,6,7,8,9, 10, 11, 12). Let r be the relation on the...
2. Let S 11,2,3,4,5, 6, 7,8,91 and let T 12,4,6,8. Let R be the relation on P (S) detined by for all X, Y E P (s), (X, Y) E R if and only if IX-T] = IY-T]. (a) Prove that R is an equivalence relation. (b) How many equivalence classes are there? Explain. (c) How mauy elements of [ø], the equivalence class of ø, are there? Explain (d) How many elements of [f1,2,3, 4)], the equivalence class of (1,2,3,...
1) Let R be the relation defined on N N as follows: (m, n)R(p, q) if and only if m - pis divisible by 3 and n - q is divisible by 5. For example, (2, 19)R(8,4). 1. Identify two elements of N X N which are related under R to (6, 45). II. Is R reflexive? Justify your answer. III. Is R symmetric? Justify your answer. IV. Is R transitive? Justify your answer. V.Is R an equivalence relation? Justify...
Q-4. [8+3+3+3+3 marks] Let be the partial order relation defined on , where means. a) Draw the Hasse diagram for . b) Find all maximal and minimal elements. c) Find lub({6,12}). a) Find glb({6,12}). e) What is the least element? The greatest element? Q-4. [8+3+3+3+3 marks] Let R be the partial order relation defined on A = {2,3, 6, 9, 10, 12, 14, 18, 20}, where xRy means x|y. a) Draw the Hasse diagram for R. b) Find all maximal...
(b) The delay time for a packet travelling between four routers (a, b, c, d) is given in the following table: To b From 1 msec 2 msec a b 2 msec 5 msec 2 msec C 5 msec 5 msec 2msec Draw the relation digraph from the information given by the above table (i) (10/100) (ii) Relation R on set {a, b, c, d} is defined as follows: "xRy if the delay time from router x to router y...
Problem 4. Let A, B e Rmxn. We say that A is equivalent to B if there exist an invertible m x m n x n matrix Q such that PAQ = B. matrix P and an invertible (a) Prove that the relation "A is equivalent to B" is reflexive, symmetric, and transitive; i.e., prove that: (i) for all A E Rmx", A is equivalent to A; (ii) for all A, B e Rmxn, if A is equivalent to B...
I. Let each of R, S, and T be binary relations on N2 as defined here: R-[<m, n EN nis the smallest prime number greater than or equal to m] S -[< m, n> EN* nis the greatest prime number less than or equal to m] (a) Which (if any) of these binary relations is a (unary) function? (b) Which (if any) of these binary relations is an injection? (c) Which (if any) of these binary relations is a surjection?...
In java please
Program 5.1: Properties of a Relation Write a program that can determine which, if any, of the following properties a binary matrix exhibits: symmetric anti-symmetric asymmetric (from the textbook) reflexive or anti-reflexive (or neither of course) Program Requirements: Hard code at least 4 binary matrices all of size 4x4. Display each binary matrix and the the properties it exhibits. For example: A 0 100 0 00 0 0000 0 000 A anti-reflexive, anti-symmetric, asymmetric B 11 10...
discrete math
Need 7c 9ab 10 15 16 17
(7) Consider the following matrices. Compute the following matrices A=[ ]B=[ 1 c-[! (a) CA (b) BAA (c) AOC (9) Determine if the following statements are True or False. If the statement is False, explain why. (a) Consider A={1,2,3,4,5). Do A1 = {1,3,5}, A2 = {2,4}. (i) Show that P ={A1, A2} forms a partition of A. (ii) Construct the matrix of the relation R corresponding to P (b) Consider A...
(b) Answer the following questions for the relation R defined on the set of seven-bit strings by s,RS2, provided that the first four bits of s, and s2 coincide. Jawab soalan-soalan berikut untuk hubungan R yang ditakrif di set rentetan tujuh-bit oleh s1Rs2, dengan syarat bahawa empat bit pertama s1 dan s2 serentak. (i) Show that R is an equivalence relation. Tunjukkan R adalah hubungan kesetaraan. (20/100) (ii) How many equivalence classes are there? Berapa banyak kelas kesetaraan yang ada?...
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