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Q29E

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Discrete Mathematics and its Applications
Found in: Page 607
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

To find the smallest relation of the relation \(\{ (1,2),(1,4),(3,3),(4,1)\} \) which is reflexive, symmetric and transitive.

The smallest relation of the relation \(\{ (1,2),(1,4),(3,3),(4,1)\} \) which is reflexive, symmetric and transitive.

\(\{ (1,1),(2,2),(3,3),(4,4),(1,2)(1,4),(2,4)(2,1),(4,2),(4,1)\} \).

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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Given data

The relation \(\{ (1,2),(1,4),(3,3),(4,1)\} \).

Step 2: Concept used of types of relation

A relation \({\rm{R}}\) on a set \({\rm{A}}\) is called reflexive if \((a,a) \in R\) for every element \(a \in A\).

A relation \({\rm{R}}\) on a set \({\rm{A}}\) is called symmetric if \((b,a) \in R\) whenever \((a,b) \in R\), for all \(a,b \in A\)

A relation \({\rm{R}}\) on a set \({\rm{A}}\) is called transitive if whenever \((a,b) \in R\) and \((b,c) \in R\) then \((a,c) \in R\) for all \(a,b,c \in A\).

Step 3: Find the smallest relation

Consider the relation \(\{ (1,2),(1,4),(3,3),(4,1)\} \) defined on set \(A = \{ 1,2,3,4\} \).

The relation

\(\{ (1,1),(2,2),(3,3),(4,4),(1,2)(1,4),(2,4)(2,1),(4,2),(4,1)\} \)

Is the smallest relation which is Reflexive.

\(\{ (1,1),(2,2),(3,3),(4,4)\} \in \{ 1,2,3,4\} \)

Symmetric,

\((1,2)(2,1) \in R\)

\((4,2),(2,4) \in R\)

and transitive relation,

\((4,1)\} (1,4) \in R\)

\((1,1),(2,2) \in R{\rm{ }}(1,2) \in R\),

\((1,2)(1,4) \in R{\rm{ }}(2,4) \in R\)

Therefore,

The smallest relation of the relation \(\{ (1,2),(1,4),(3,3),(4,1)\} \) which is reflexive, symmetric and transitive is \(\{ (1,1),(2,2),(3,3),(4,4),(1,2)(1,4),(2,4)(2,1),(4,2),(4,1)\} \).

Most popular questions for Math Textbooks

Show that the relation R=ϕ on a non-empty set S=ϕ is symmetric, transitive and reflexive.

Show that if \(C\) is a condition that elements of the \(n\)-ary relation \(R\)and \(S\)may satisfy, then \({s_C}(R - S) = {s_C}(R) - {s_C}(S)\).

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

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 \(R^\circ S\)?

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.

34. Find

(a) \({R_1} \cup {R_3}\).

(b) \({R_1} \cup {R_5}\).

(c) \({R_2} \cap {R_4}\).

(d) \({R_3} \cap {R_5}\).

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

(f) \({R_2} - {R_1}\).

(g) \({R_1} \oplus {R_3}\).

(h) \({R_2} \oplus {R_4}\).
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