Question

To determine the list of ordered pairs in the equivalence relations produced by these partitions of \(\{ 0,1,2,3,4,5\} \).

Short answer

The list of ordered pair is \(R = \{ (0,0),(1,1),(2,2),(3,3),(4,4),(5,5)\} \).

Step-by-step solution

Given data

The partition is \(\{ 0\} ,\{ 1\} ,\{ 2\} ,\{ 4\} ,\{ 5\} \).

Concept used of partition of sets

Let\(a\)and\(b\)are in the same subset of the partition if and only if\((a,b) \in R\).

Find partition of set

The partition is \(\{ 0\} ,\{ 1\} ,\{ 2\} ,\{ 4\} ,\{ 5\} \).

Assume that, \(R\) be the equivalence relation produced by the partition \(\{ \{ 0\} ,\{ 1\} ,\{ 2\} ,\{ 4\} ,\{ 5\} \} \) of \(\{ 0,1,2,3,4,5\} \).

The list of ordered pairs is,

\(R = \{ (0,0),(1,1),(2,2),(3,3),(4,4),(5,5)\} \)

Hence, the list of ordered pairs is \(R = \{ (0,0),(1,1),(2,2),(3,3),(4,4),(5,5)\} \).