Let A 013 that is A is the set of all ordered triples with

Let A = {0,1}3: that is, A is the set of all ordered triples with entries from 0 and 1. Then define a relation R A×A such that xRy if and only if x and y have the same number of 0s. Note that R is an equivalence relation.

Give the partition of A created by the equivalence classes of R.

Solution

1.

xRx trivially

SO, R is reflexive

2. Let, xRy ie x and y have same number of 0s so yRx

3. Let, xRy and yRz

So, x and y and z have same number of 0s

So, xRz

So, R is transitive anr symmetric and hence equivalence relation

Four partitions

[(1,1,1)] = Equivalence class with no 0s

[(0,1,1)] = Equivalence class iwht 1 0s

[(0,0,1)]= Equivalence class with 2 0s

[(0,0,0)] =Equivalence class iwth 3 0s