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Q21E

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Discrete Mathematics and its Applications
Found in: Page 582
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

Determine whether the relation R on the set of all real numbers are asymmetric.

The given relation is not asymmetric.

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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Given data

x+y=0

Step 2: Concept used of relation

A relation R on a set A is called reflexive if (a,a)∈R for every element a∈A.

A relation R on a set A is called symmetric if (b,a)∈R whenever (a,b)∈R, for all a,b∈A

A relationR on a set A such that for all a,b∈A, if(a,b)∈R and (b,a)∈R then a=b is called anti symmetric.

A relation R on a set A is called transitive if whenever (a,b)∈Rand (b,c)∈R then (a,c)∈R for all a,b,c∈A

Step 3: Solve for relation

An asymmetric relation is one for which (a,b)∈R and (b,a)∈R can never hold simultaneously, even if a=b.

Thus R is asymmetric if and only if R is anti symmetric and also irreflexive.

(x,y)∈R

x+y=0

y+x=0

(y,x)∈R

Thus, R is not asymmetric.

Most popular questions for Math Textbooks

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

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.

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

In Exercises 25–27 list all ordered pairs in the partial ordering with the accompanying Hasse diagram. 26.

Can a relation on a set be neither reflexive nor irreflexive?
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