
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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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) 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.](https://content.bartleby.com/qna-images/question/a040b373-32a4-4e5c-bccb-49a589c6504e/94d4e36b-d4fb-44fd-90ac-55d52576f729/ytu83x4g_thumbnail.png)
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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- How many relations are there on the set {a, b}. Show all possible relations on the set {a, b} in matrix format.arrow_forwardDefine: what is an equivalence relation?arrow_forwardDetermine whether the given relation is an equivalent relation on (1, 2, 3, 4, 5} If the relation is an equivalence relation, list the equivalence classes (x, y E {1,2, 3, 4, 5} . ) 3, 4, 5}. 1. {(1,1), (2, 2) , (3, 3) , (4, 4) , (5, 5) , (1, 3) , (3, 1)} 2. {(x, y) | 3 divides x + y}arrow_forward
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