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Chapter 9: Q6E (page 606)

In Exercises 5-7 draw the directed graph of the reflexive closure of the relations with the directed graph shown.

6.

Short Answer

Expert verified

The directed graph of the reflexive closure of the relation is then

loops added at every vertex in the given directed graph.

Step by step solution

01

Given

The reflexive closure of\(R\)is\(R \cup \Delta = R \cup \{ (a,a)\mid a \in A\} \)where\(\{ (a,a)\mid a \in A\} \)represents loops at every vertex in the directed graph.

02

Concept of Reflexive Closure

The reflexive closure of\(R\) is the relation that contains all ordered pairs of\(R\)and to which all ordered pairs of the form\((a,a) \in R(a \in A)\)were added (when they were not present yet).

\(R \cup \Delta = R \cup \{ (a,a)\mid a \in A\} \).

03

Find the Reflexive Closure

The reflexive closure of\(R\)is\(R \cup \Delta = R \cup \{ (a,a)\mid a \in A\} \)where\(\{ (a,a)\mid a \in A\} \)represents loops at every vertex in the directed graph.

The directed graph of the reflexive closure of the relation is then

loops added at every vertex in the given directed graph.

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Most popular questions from this chapter

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

Must an asymmetric relation also be antisymmetric? Must an antisymmetric relation be asymmetric? Give reasons for your answers.

To find the ordered pairs \((a,b)\) in \({R^2}\;\& \;\;{R^n}\) relation where \(n\) is a positive integer.

To find the transitive closers of the relation \(\{ (1,2),(2,1),(2,3),(3,4),(4,1)\} \) with the use of Warshall’s algorithm.

To prove the closure with respect to the property. ofthe relation \(R = \{ (0,0),(0,1),(1,1),(2,2)\} \) on the set \(\{ 0,1,2\} \) does not exist if . is the property" has an odd number of elements."
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