If {{a, c, e}, {b, d, f}} is a partition of the set A = {a, b, c, d, e, f}, determine the correspondingequivalence relation R.

An equivalence relation R corresponding to a partition of a set divides the set into mutually exclusive and collectively exhaustive subsets, where each pair of elements within the same subset are related.

Given the partition {{a, c, e}, {b, d, f}} of the set A = {a, b, c, d, e, f}, we can determine the corresponding equivalence relation R as follows:

Step 1: Identify the subsets in the partition. The subsets are {a, c, e} and {b, d, f}.

Step 2: Form pairs of related elements within each subset. In the subset {a, c, e}, the pairs of related elements are (a, a), (a, c), (a, e), (c, a), (c, c), (c, e), (e, a), (e, c), and (e, e). In the subset {b, d, f}, the pairs of related elements are (b, b), (b, d), (b, f), (d, b), (d, d), (d, f), (f, b), (f, d), and (f, f).

Step 3: Combine all pairs of related elements to form the equivalence relation R. So, the equivalence relation R corresponding to the given partition is: R = {(a, a), (a, c), (a, e), (c, a), (c, c), (c, e), (e, a), (e, c), (e, e), (b, b), (b, d), (b, f), (d, b), (d, d), (d, f), (f, b), (f, d), (f, f)}.

This means that any two elements in the same subset of the partition are related, and there is no relation between elements in different subsets.