Question

To find the ordered pairs in \({R^3}\) relation.

Short answer

The ordered pairs in \({R^3}\) is \(\{ (a,b)\mid a\) is a grand grand parent of \(b\} \).

Step-by-step solution

 Given

It is given that \(R\) is the parent relation on the set of all the people.

The Concept of ordered pairs in relation

A relation is any collection of ordered pairs. The fact that the pairs are ordered is important, and means that the ordered pair (a, b) is different from the ordered pair (b, a) unless a = b. For most useful relations, the elements of the ordered pairs are naturally associated or related in some way.

Determine the relation

Let us find the relation in the ordered pair\({R^3}\).

Let\(A\)be the set of all people and\(R\)be a relation defined on\(A\)by

\((a,b) \in R\)if and only if\(a\)is a parent of\(b\).

Let us find the relation in the ordered pair\({R^3}\).

Let\(A\)be the set of all people and\(R\)be a relation defined on\(A\)by

\((a,b) \in R\)if and only if\(a\)is a parent of\(b\).

Then,

\(\left\{ {(a,b) \in {R^2}/ \leftrightarrow b} \right.\)such that\((a,b) \in R\)and\(\left. {(b,c) \in R} \right\}\)

There is a person\(b\)such that\(a\)is a parent of\(b\)and\(b\)is a parent of\(c\).

\(a\)is a grandparent of\(c\).

Now,

\(\left\{ {(a,b) \in {R^3}/ \leftrightarrow b} \right.\)such that\((a,b) \in {R^2}\)and\(\left. {(b,c) \in R} \right\}\)

There is a person \(b\) such that \(a\) is a grandparent of \(a\) and \(b\) is a parent of \(c\).

\(a\)is a grandparent of\(c\).

Hence,

\({R^3} = \{ (a,b)/a\)is a grand grand parent of\(b\} \)

Hence the ordered pairs of\({R^3} = \{ (a,b)\mid a\)is a grand grand parent of\(b\} \)is described.

Conclusion:

Hence the ordered pairs in \({R^3}\) is \(\{ (a,b)\mid a\)is a grand grand parent of \(b\} \).