# Partitions of a power set and equivalence classes

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?

• Yes. You can test for yourself that this is an equivalence relation by finding if it is reflexive, symmetric, and transitive.
– anak
Jul 24, 2015 at 15:33

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

• Exactly. Would there be another way to define the relation, so I would get the desired classes? Jul 24, 2015 at 15:36
• Well, anything which you can express formally and which means (logically) the same thing is OK. I cannot think of another way right now but there may be another way. Jul 24, 2015 at 15:42
• OK. Thank you very much! Jul 24, 2015 at 15:51