Question

To find the ordered pairs \((a,b)\) in \({R^2}\;\& \;\;{R^n}\) relation where \(n\) is a positive integer.

Short answer

The ordered pairs in \((a,b)\) in \({R^2}\;\& \;\;{R^n}\) relation is as shown.

Step-by-step solution

 Step 1:  Given

It is given that \((a,b) \in R\) if and only if \(a\) was the thesis advisor of \(b\).

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\({R^2}\;\& \;\;{R^n}\).

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

\((a,b) \in R\)if and only\(a\)was the thesis advisor of\(b\).

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

There is a person\(b\)with doctorate such that\(a\)was the thesis advisor of\(b\)and\(b\)was the thesis advisor of\(c\).

Hence, \(\left\{ {(a,b) \in {R^2} \leftrightarrow {x_1}} \right.\) in A such that is the advisor of and is the advisor of \(b\)

For any positive integer\(n\),

\((a,b) \in {R^n} \leftrightarrow {x_1},{x_2}, \ldots ,{x_{n - 1}}\)in\(A\)such that\(a\)is the advisor of\({x_1},{x_{n - 1}}\)is the advisor of\(b\)and\({x_1}\)is the advisor of\({x_{i + 1}}\forall 1 \le i \le n - 2\).

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

Conclusion:

Hence the ordered pairs in \((a,b)\) in \({R^2}\;\& \;\;{R^n}\) relation is as shown.