StudySmarter AI is coming soon!

- :00Days
- :00Hours
- :00Mins
- 00Seconds

A new era for learning is coming soonSign up for free

Suggested languages for you:

Americas

Europe

Q29E

Expert-verifiedFound in: Page 607

Book edition
7th

Author(s)
Kenneth H. Rosen

Pages
808 pages

ISBN
9780073383095

**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.

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

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

**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 \(a,b,c \in A\)****.**

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

94% of StudySmarter users get better grades.

Sign up for free