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Expert-verified Found in: Page 582 ### Discrete Mathematics and its Applications

Book edition 7th
Author(s) Kenneth H. Rosen
Pages 808 pages
ISBN 9780073383095 # Let $$R$$be the relation on the set of people consisting of pairs $$(a,b)$$, where $$a$$ is a parent of $$b$$. Let $$S$$ be the relation on the set of people consisting of pairs $$(a,b)$$, where $$a$$ and $$b$$are siblings (brothers or sisters). What are $$S^\circ R$$ and $$R^\circ S$$?

$$S^\circ R = \{ (a,b)\mid {\rm{ a is a parent of }}b\}$$

$$R^\circ S = \{ (a,b)\mid$$a is an uncle or aunt of $$b\}$$

See the step by step solution

## Step 1: Given Data

Let $$R$$be the relation on the set of people consisting of pairs $$(a,b)$$, where $$a$$is a parent of $$b$$. Let $$S$$ be the relation on the set of people consisting of pairs $$(a,b)$$, where $$a$$ and $$b$$ are siblings (brothers or sisters).

## Step 2: Concept of the composite relation

Let $$R$$ be a relation from a set $$A$$ to a set $$B$$and $$S$$ a relation from $$B$$ to a set $$C$$. The composite of $$R$$and $$S$$ is the relation consisting of ordered pairs $$(a,c)$$, where $$a \in A,c \in C$$, and for which there exists an element $$b \in B$$ such that $$(a,b) \in R$$ and $$(b,c) \in S$$. We denote the composite of $$R$$and $$S$$ by $$S^\circ R$$.

## Step 3: Determine the value $$S^\circ R$$ and $$R^\circ S$$

Consider, $$(a,b) \in S^\circ R$$, this means there is $$c$$ such that $$(a,c) \in R$$and $$(c,b) \in S$$, or in other words, $$a$$ is a parent of $$c$$ as well as $$c$$ and $$b$$ are siblings. This implies $$a$$ is a parent of $$b$$ too.

$$S^\circ R = \{ (a,b)\mid$$ a is a parent of $$b\}$$

Similarly, if $$(a,b) \in R^\circ S$$, then there is $$c$$ so that $$(a,c) \in S$$, i.e. $$a$$ and $$c$$ are siblings and $$(c,b) \in R$$, or $$c$$ is a parent of $$b$$. Thus $$a$$ is an uncle or aunt of $$b$$.

$$R^\circ S = \{ (a,b)\mid$$ a is an uncle or aunt of $$b\}$$.

Therefore, $$S^\circ R = \{ (a,b)\mid {\rm{ a is a parent of }}b\}$$

$$R^\circ S = \{ (a,b)\mid$$a is an uncle or aunt of $$b\}$$ ### Want to see more solutions like these? 