Question

Let A = {1, 2, 3, 4, 5, 6} Which of the following partitions of A correspond to an equivalence relation on A?

Options

• {1, 3}, {2, 4, 5}, {6}

• {1, 2, }, {3, 4}, {2, 3, 5, 6}

• none of these

• {1, 2, 3}, {3, 4, 5, 6}

Answer 1

{1, 3}, {2, 4, 5}, {6}

Explanation:

For the partition sub-sets to be an equivalence relation, the following conditions must be met:

The partition sub-sets must be distinct, meaning they must not share any entries. Their sum must equal that of the main set (super-set) The partition sub-sets {1,3},{2,4},{5,6} are pairwise disjoint in the set A ={1,2,3,4,5,6} and their union, i.e. {1,3} U {2,4,5} U {6} is pairwise disjoint in the set A = {1,2,3,4,5,6}. The criterion for the partition sub-sets to be an equivalence relation of the set A is U {6} = {1,2,3,4,5,6} = A.