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Found in: Page 607

### Discrete Mathematics and its Applications

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.

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

See the step by step solution

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

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