[Math] Partitions of a power set and equivalence classes

Question

I got the set:

$$
M=\{1,2,3,4\}.
$$

I could split the power set of M into the following subsets:

$$
P_{0}=\{\emptyset\} \\
P_{1}=\{\{1\},\{2\},\{3\},\{4\}\} \\
P_{2}=\{\{1,2\},\{1,3\},\{1,4\},\{2,3\},\{2,4\},\{3,4\}\} \\
P_{3}=\{\{1,2,3\},\{1,2,4\},\{1,3,4\},\{2,3,4\}\} \\
P_{4}=\{\{1,2,3,4\}\}
$$

The set of those subsets would give me a partition of the power set:

$$
P=\{P_{0},P_{1},P_{2},P_{3},P_{4}\} \\
$$

We can interpret the partition as a set of equivalence classes. The equvalence relation would be then defined as:

$$
aRb :\Leftrightarrow |a|=|b|
$$

Is this correct?

Answer 1

Yes, that's correct. By your definition two elements a and b (elements i.e. subsets) are equivalent if their cardinality is the same.