Book edition 7th

Author(s) Kenneth H. Rosen

Pages 808 pages

ISBN 9780073383095

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Short Answer

Show that the relation $R = ϕ$ on a non-empty set $S$ is symmetric and transitive, but not reflexive.

Hence $R$ is symmetric, transitive and not reflexive.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Given data

The relation $R = ϕ$ on a non-empty set is given.

Step 2: Concept used of relation

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

A relation $R$ on a set $A$ is called symmetric if $(b, a) \in R$ whenever $(a, b) \in R$ , for all $a, b \in A$ .

A relation $R$ on a set $A$ such that for all $a, b \in A$ , if $(a, b) \in R$ and $(b, a) \in R$ then $a = b$ is called anti symmetric.

A relation $R$ on a set $A$ 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: Solve for relation

Symmetric Since $(a, b) \in R$ is always false (as $R$ is the empty set), the conditional statement $((a, b) \in R) \to B$ is true for any statement B.

Let B be the statement " $(b, a) \in R$ ". The conditional statement

"If $(a, b) \in R$ , then $(b, a) \in R$ " is then always true and thus the relation $R$ is symmetric by the definition of symmetric. Transitive Since $(a, b) \in R$ is always false (as $R$ is the empty set) and since $(b, a) \in R$ is always false, the statement " $(a, b) \in R$ , and $(b, a) \in R$ " is also always false. Then the conditional statement $((a, b) \in R$ and $(b, a) \in R) \to B$ is true for any statement B.

Let B the statement " $a = b$ ". The conditional statement

" If $(a, b) \in R$ and $(b, a) \in R$ , then $a = b$ is then always true and thus the relation $R$ is transitive by the definition of transitive.

Not reflexive For any element $a \in S : (a, a) \notin R$ because $R$ is the empty set. By the definition of reflexive, $R$ is then not reflexive.

Hence $R$ is symmetric, transitive and not reflexive.

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Discrete Mathematics and its Applications

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Short Answer

Show that the relation $R = ϕ$ on a non-empty set $S$ is symmetric and transitive, but not reflexive.

Step by Step Solution

Step 1: Given data

Step 2: Concept used of relation

Step 3: Solve for relation

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Discrete Mathematics and its Applications

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Short Answer

Show that the relation R=ϕ on a non-empty set Sis symmetric and transitive, but not reflexive.

Step by Step Solution

Step 1: Given data

Step 2: Concept used of relation

Step 3: Solve for relation

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Show that the relation $R = ϕ$ on a non-empty set $S$ is symmetric and transitive, but not reflexive.