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Q33E

Expert-verifiedFound in: Page 582

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 \(b\} \)

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).

**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 \(S^\circ R\).**

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 \(b\} \)

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