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? 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)
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? 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
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.7: Relations
Problem 6TFE: Label each of the following statements as either true or false. Let R be a relation on a nonempty...
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Question
![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
ro 3
3 0
0 1 1
L2 1 2 0.
b.Draw the graph with the adjacency matrix
with respect to the
2
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.
e
a
de
b
og
h
i. Find the maximal elements.
ii. Find the minimal elements.
iii. Is there a greatest element?
iv. Is there a least element?
v. Find all upper bounds of {m, k, s}.
vi. Find all lower bounds of {c, d, t}.
vii. Find the greatest lower bound of fu, k, m} if it exists.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa040b373-32a4-4e5c-bccb-49a589c6504e%2F94d4e36b-d4fb-44fd-90ac-55d52576f729%2Fytu83x4g_processed.png&w=3840&q=75)
Transcribed Image Text: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
ro 3
3 0
0 1 1
L2 1 2 0.
b.Draw the graph with the adjacency matrix
with respect to the
2
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.
e
a
de
b
og
h
i. Find the maximal elements.
ii. Find the minimal elements.
iii. Is there a greatest element?
iv. Is there a least element?
v. Find all upper bounds of {m, k, s}.
vi. Find all lower bounds of {c, d, t}.
vii. Find the greatest lower bound of fu, k, m} if it exists.
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