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2. Let R be the relation on the set A = (1, 2, 3, 4, 5, 6, 7} defined by the rule (a,b) e R if the integer (a- b) is divisible by 4. List the elements of R and its inverse? b) Check whether the relation R on the set S = (1, 2, 3} is an equivalent relation where R = {(1,1), (2,2), (3,3), (2,1), (1,2), (2,3), (1,3), (3,1)}. Which of the following properties R has: reflexive, symmetric, anti-symmetric, transitive? Justify your answer in each case? Let S = {a, b, c} and R = {(a,a), (b, b), (c,c), (b, c), (c, b)}, find [a], (b] and [c] (that is the equivalent class of a, b, and c). Hence or otherwise find the set of equivalent class of a, b and c? c)

2. Let R be the relation on the set A = (1, 2, 3, 4, 5, 6, 7} defined by the rule (a,b) e R if the integer (a- b) is divisible by 4. List the elements of R and its inverse? b) Check whether the relation R on the set S = (1, 2, 3} is an equivalent relation where R = {(1,1), (2,2), (3,3), (2,1), (1,2), (2,3), (1,3), (3,1)}. Which of the following properties R has: reflexive, symmetric, anti-symmetric, transitive? Justify your answer in each case? Let S = {a, b, c} and R = {(a,a), (b, b), (c,c), (b, c), (c, b)}, find [a], (b] and [c] (that is the equivalent class of a, b, and c). Hence or otherwise find the set of equivalent class of a, b and c? c)

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter8: Polynomials
Section8.1: Polynomials Over A Ring
Problem 25E: (See exercise 24.) Show that the relation f(x)Rg(x) if and only if f(x)=g(x) is an equivalence...
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iii. Find the inverse of the function f c. The functions h and m are defined on the set R of real numbers by h: x→ax +b and m: x → cx -d, where a, b, c and d are constants. If h om = moh, show that (c-1)b = (a-1)d 2. Let R be the relation on the set A = {1, 2, 3, 4, 5, 6, 7} defined by the rule (a,b) eR if the integer (a- b) is divisible by 4. List the elements of R and its inverse? b) Check whether the relation R on the set S = {1, 2, 3} is an equivalent relation where R = {(1,1), (2,2), (3,3), (2,1), (1,2), (2,3), (1,3), (3,1)}. Which of the following properties R has: reflexive, symmetric, anti-symmetric, transitive? Justify your answer in each case? c) Let S = {a, b, c} and R = {(a, a), (b, b), (c, c), (b, c), (c, b)}, find [a], [b] and [c] (that is the equivalent class of a, b, and c). Hence or otherwise find the set of equivalent class of a. b and c? 3. a. Define the following with an example; i. paths ii. simple graph гО 3 21 3 0 1 b.Draw the graph with the adjacency matrix with respect to the 0 1 1 2 L2 1 2 0 ordering of vertices, a, b, c, d. i. Find the degree of each vertex in your graph from part (a) above. ii. How many walks of length 2 are there from the vertex c to c? How many of these walks are paths? 4. a. Define the following Terms giving one example each: i. Partial Ordering Relations ii. Equivalence relations b. Answer these questions for the partial order represented by the following Hasse diagram. a d' m b• g P h i. Find the maximal elements. ii. Find the minimal elements. ii. Is there a greatest element? iv. Is there a least element?
Transcribed Image Text:iii. Find the inverse of the function f c. The functions h and m are defined on the set R of real numbers by h: x→ax +b and m: x → cx -d, where a, b, c and d are constants. If h om = moh, show that (c-1)b = (a-1)d 2. Let R be the relation on the set A = {1, 2, 3, 4, 5, 6, 7} defined by the rule (a,b) eR if the integer (a- b) is divisible by 4. List the elements of R and its inverse? b) Check whether the relation R on the set S = {1, 2, 3} is an equivalent relation where R = {(1,1), (2,2), (3,3), (2,1), (1,2), (2,3), (1,3), (3,1)}. Which of the following properties R has: reflexive, symmetric, anti-symmetric, transitive? Justify your answer in each case? c) Let S = {a, b, c} and R = {(a, a), (b, b), (c, c), (b, c), (c, b)}, find [a], [b] and [c] (that is the equivalent class of a, b, and c). Hence or otherwise find the set of equivalent class of a. b and c? 3. a. Define the following with an example; i. paths ii. simple graph гО 3 21 3 0 1 b.Draw the graph with the adjacency matrix with respect to the 0 1 1 2 L2 1 2 0 ordering of vertices, a, b, c, d. i. Find the degree of each vertex in your graph from part (a) above. ii. How many walks of length 2 are there from the vertex c to c? How many of these walks are paths? 4. a. Define the following Terms giving one example each: i. Partial Ordering Relations ii. Equivalence relations b. Answer these questions for the partial order represented by the following Hasse diagram. a d' m b• g P h i. Find the maximal elements. ii. Find the minimal elements. ii. Is there a greatest element? iv. Is there a least element?
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