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Q33E

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Discrete Mathematics and its Applications

Found in: Page 582

Book edition 7th

Author(s) Kenneth H. Rosen

Pages 808 pages

ISBN 9780073383095

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Short Answer

Let \(R\)be the relation on the set of people consisting of pairs \((a,b)\), where \(a\) is a parent of \(b\). Let \(S\) be the relation on the set of people consisting of pairs \((a,b)\), where \(a\) and \(b\)are siblings (brothers or sisters). What are \(S^\circ R\) and \(R^\circ S\)?

\(S^\circ R = \{ (a,b)\mid {\rm{ a is a parent of }}b\} \)

\(R^\circ S = \{ (a,b)\mid \)a is an uncle or aunt of \(b\} \)

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Given Data

Let \(R\)be the relation on the set of people consisting of pairs \((a,b)\), where \(a\)is a parent of \(b\). Let \(S\) be the relation on the set of people consisting of pairs \((a,b)\), where \(a\) and \(b\) are siblings (brothers or sisters).

Step 2: Concept of the composite relation

Let \(R\) be a relation from a set \(A\) to a set \(B\)and \(S\) a relation from \(B\) to a set \(C\). The composite of \(R\)and \(S\) is the relation consisting of ordered pairs \((a,c)\), where \(a \in A,c \in C\), and for which there exists an element \(b \in B\) such that \((a,b) \in R\) and \((b,c) \in S\). We denote the composite of \(R\)and \(S\) by \(S^\circ R\).

Step 3: Determine the value \(S^\circ R\) and \(R^\circ S\)

Consider, \((a,b) \in S^\circ R\), this means there is \(c\) such that \((a,c) \in R\)and \((c,b) \in S\), or in other words, \(a\) is a parent of \(c\) as well as \(c\) and \(b\) are siblings. This implies \(a\) is a parent of \(b\) too.

\(S^\circ R = \{ (a,b)\mid \) a is a parent of \(b\} \)

Similarly, if \((a,b) \in R^\circ S\), then there is \(c\) so that \((a,c) \in S\), i.e. \(a\) and \(c\) are siblings and \((c,b) \in R\), or \(c\) is a parent of \(b\). Thus \(a\) is an uncle or aunt of \(b\).

\(R^\circ S = \{ (a,b)\mid \) a is an uncle or aunt of \(b\} \).

Therefore, \(S^\circ R = \{ (a,b)\mid {\rm{ a is a parent of }}b\} \)

\(R^\circ S = \{ (a,b)\mid \)a is an uncle or aunt of \(b\} \)

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Exercises 34–37 deal with these relations on the set of real numbers:

\({R_1} = \left\{ {\left( {a,\;b} \right) \in {R^2}|a > b} \right\},\) the “greater than” relation,

\({R_2} = \left\{ {\left( {a,\;b} \right) \in {R^2}|a \ge b} \right\},\) the “greater than or equal to” relation,

\({R_3} = \left\{ {\left( {a,\;b} \right) \in {R^2}|a < b} \right\},\) the “less than” relation,

\({R_4} = \left\{ {\left( {a,\;b} \right) \in {R^2}|a \le b} \right\},\) the “less than or equal to” relation,

\({R_5} = \left\{ {\left( {a,\;b} \right) \in {R^2}|a = b} \right\},\) the “equal to” relation,

\({R_6} = \left\{ {\left( {a,\;b} \right) \in {R^2}|a \ne b} \right\},\) the “unequal to” relation.

35. Find

(a) \({R_2} \cup {R_4}\).

(b) \({R_3} \cup {R_6}\).

(c) \({R_3} \cap {R_6}\).

(d) \({R_4} \cap {R_6}\).

(e) \({R_3} - {R_6}\).

(f) \({R_6} - {R_3}\).

(g) \({R_2} \oplus {R_6}\).

(h) \({R_3} \oplus {R_5}\).

How many transitive relations are there on a set with \(n\) elements if

a) \(n = 1\) ?

b) \(n = 2\) ?

c) \(n = 3\) ?

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