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Q21E

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Discrete Mathematics and its Applications
Found in: Page 631
Discrete Mathematics and its Applications

Discrete Mathematics and its Applications

Book edition 7th
Author(s) Kenneth H. Rosen
Pages 808 pages
ISBN 9780073383095

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Short Answer

Draw the Hasse diagram for the less than or equal to relation on \(\{ 0,2,5,10,11,15\} \).

The Hasse diagram for \(\{ (0,2,5,10,11,15), \le \} \)

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Given data

Given data is \((0,2,5,10,11,15)\).

Step 2: Concept used Hasse diagram

Hasse diagrams are obtained from a directed graph of a partial ordering by,

1) Removing all loops due to reflexivity from the graph of a partial ordering.

2) Removing all edges occurring due to transitivity of the partial ordering.

3) Arranging all edges to point upwards and deleting (directional) arrows

Thus to get all ordered pairs ordered pairs in the partial ordering for a given Hasse diagram, we look for pairs \((x,y)\) such that path from \(x\) to \(y\) is going upwards . In addition, we also need to add pairs of the form \((x,x)\) to account for reflexive pairs (loops).

Step 3: Draw the Hasse diagram

Consider greater than or equal to relation on \(\{ 0,2,5,10,11,15\} \) which is represented as \(\{ (0,2,5,10,11,15), \le \} \). The Hasse diagram for \(\{ (0,2,5,10,11,15), \le \} \)

Hence, Hasse diagram for less than or equal to relation on \(\{ 0,2,5,10,11,15\} \) is drawn.

Most popular questions for Math Textbooks

Which relations in Exercise 5 are asymmetric?

What is the covering relation of the partial ordering \(\{ (a,b)\mid a\) divides \(b\} \) on \(\{ 1,2,3,4,6,12\} \).

Suppose that \(R\) and \(S\) are reflexive relations on a set \(A\).

Prove or disprove each of these statements.

a) \(R \cup S\) is reflexive.

b) \(R \cap S\) is reflexive.

c) \(R \oplus S\) is irreflexive.

d) \(R - S\) is irreflexive.

e) \(S^\circ R\) is reflexive.

Show that if \(R\) and \(S\) are both \(n\)-ary relations, then

\({P_{{i_1},{i_2}, \ldots ,{i_m}}}(R \cup S) = {P_{{i_1},{i_2}, \ldots ,{i_m}}}(R) \cup {P_{{i_1},{i_2}, \ldots ,{i_m}}}(S)\).

Let \(A\) be the set of students at your school and \(B\) the set of books in the school library. Let \({R_1}\) and \({R_2}\) be the relations consisting of all ordered pairs \((a,b)\), where student \(a\) is required to read book \(b\) in a course, and where student \(a\) has read book \(b\), respectively. Describe the ordered pairs in each of these relations.

a) \({R_1} \cup {R_2}\)

b) \({R_1} \cap {R_2}\)

c) \({R_1} \oplus {R_2}\)

d) \({R_1} - {R_2}\)

e) \({R_2} - {R_1}\)
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