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 itsinverse?b)Check whether the relation R on the set S = {1, 2, 3} is an equivalentrelation whereR = {(1,1), (2,2), (3,3), (2,1), (1,2), (2,3), (1,3), (3,1)}. Which of thefollowing 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 theset of equivalent class of a, b and c?3. a. Define the following with an example;i. pathsii. simple graphro 33 00 1 1L2 1 2 0.b.Draw the graph with the adjacency matrixwith respect to the2ordering 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 manyof these walks are paths?4. a. Define the following Terms giving one example each:i. Partial Ordering Relationsii. Equivalence relationsb. Answer these questions for the partial order represented by the following Hassediagram.eadeboghi. 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.

Given: A={1, 2, 3, 4, 5, 6, 7} and R is the relation on this set defined s: (a, b)ϵR of the integer…

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)

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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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.

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