Question

(a) To find Relation\({R^2}\)

(b) To find Relation \({R^3}\)

(c) To find Relation \({R^4}\)

(d) To find Relation\({R^5}\)

Short answer

(a)The solution of Relation\({R^2} = \left\{ \begin{array}{l}(1,1),(1,2),(1,3),(1,4),(1,5),\\(2,1),(2,2),(2,4)(2,5),\\(3,1),(3,2),(3,3),(3,4),(3,5),\\(4,1),(4,2)(4,3),(4,4),\\(5,1),(5,2),(5,3),(5,4),(5,5)\end{array} \right\}\)

(b) The solution of Relation\({R^3} = \left\{ \begin{array}{l}(1,1),(1,2),(1,3),(1,4),(1,5),\\(2,1),(2,2),(2,3),(2,4),(2,5),\\(3,1),(3,2),(3,3),(3,4),(3,5),\\(4,1),(4,2),(4,3),(4,4),(4,5),\\(5,1),(5,2),(5,3),(5,4),(5,5)\end{array} \right\}\)

(c) The solution of Relation\({R^4} = \left\{ \begin{array}{l}(1,1),(1,2),(1,3),(1,4),(1,5),\\(2,1),(2,2),(2,3),(2,4)(2,5),\\(3,1),(3,2),(3,3),(3,4),(3,5),\\(4,1),(4,2)(4,3),(4,4),(4,5),\\(5,1),(5,2),(5,3),(5,4),(5,5)\end{array} \right\}\)

(d) The solution of Relation\({R^4} = \left\{ \begin{array}{l}(1,1),(1,2),(1,3),(1,4),(1,5),\\(2,1),(2,2),(2,3),(2,4)(2,5),\\(3,1),(3,2),(3,3),(3,4),(3,5),\\(4,1),(4,2)(4,3),(4,4),(4,5),\\(5,1),(5,2),(5,3),(5,4),(5,5)\end{array} \right\}\)

Step-by-step solution

 Given

(a)Relation \(R = \{ (1,1),(1,2),(1,3),(2,3),(2,4),(3,1),(3,4),(3,5),(4,2),(4,5),(5,1)\),

\(\qquad (5,2),(5,4)\} \)

On set \(A = \{ 1,2,3,4,5\} \)

(b)Relation\(R = \{ (1,1),(1,2),(1,3),(2,3),(2,4),(3,1),(3,4),(3,5),(4,2),(4,5),(5,1)\)\(\qquad (5,2),(5,4)\} \)

On set \(A = \{ 1,2,3,4,5\} \).

(c)Relation\(R = \{ (1,1),(1,2),(1,3),(2,3),(2,4),(3,1),(3,4),(3,5),(4,2),(4,5),(5,1)\)\(\qquad (5,2),(5,4)\} \)

On set \(A = \{ 1,2,3,4,5\} \).

(d)Relation\(R = \{ (1,1),(1,2),(1,3),(2,3),(2,4),(3,1),(3,4),(3,5),(4,2),(4,5),(5,1)\)\(\qquad (5,2),(5,4)\} \)

On set \(A = \{ 1,2,3,4,5\} \).

The Concept ofrelation

An n-array relation on n sets, is any subset of Cartesian product of the n sets (i.e., a collection ofn-tuples), with the most common one being a binary relation, a collection of order pairs from two sets containing an object from each set. The relation is homogeneous when it is formed with one set.

Determine the value of relation (a)

Consider the relation\(R\)

\({R^2} \Rightarrow \)paths of length 2

\(\begin{array}{c}{R^2} = R \cdot R\\ = \{ (1,1),(1,2),(1,3),(1,4),(1,5),\\(2,1),(2,2),(2,4),(2,5),\\(3,1),(3,2),(3,3),(3,4),(3,5),\\(4,1),(4,2),(4,3),(4,4),\\(5,1),(5,2),(5,3),(5,4),(5,5)\} \end{array}\)

Determine the value of relation (b)

Consider the relation \(R\)

\({R^3} \Rightarrow \)paths of length 3

\(\begin{array}{c}{R^3} = {R^2}.R\\ = \{ (1,1),(1,2),(1,3),(1,4),(1,5),\\(2,1),(2,2),(2,3),(2,4)(2,5),\\(3,1),(3,2),(3,3),(3,4),(3,5),\\(4,1),(4,2)(4,3),(4,4),(4,5),\\(5,1),(5,2),(5,3),(5,4),(5,5)\} \end{array}\)

Determine the value of relation (c)

Consider the relation\(R\)

\({R^4} \Rightarrow \)paths of length 4

\(\begin{array}{c}{R^4} = {R^3} \cdot R\\ = \{ (1,1),(1,2),(1,3),(1,4),(1,5),\\(2,1),(2,2),(2,3),(2,4)(2,5),\\(3,1),(3,2),(3,3),(3,4),(3,5),\\(4,1),(4,2)(4,3),(4,4),(4,5),\\(5,1),(5,2),(5,3),(5,4),(5,5)\} \end{array}\)

Determine the value of relation (d)

Consider the relation\(R\)

\({R^5} \Rightarrow \)paths of length 5

\(\begin{array}{c}{R^5} = {R^4} \bullet R\\ = \{ (1,1),(1,2),(1,3),(1,4),(1,5),\\(2,1),(2,2),(2,3),(2,4),(2,5),\\(3,1),(3,2),(3,3),(3,4),(3,5),\\(4,1),(4,2),(4,3),(4,4),(4,5),\\(5,1),(5,2),(5,3),(5,4),(5,5)\} \end{array}\)