Consider the equivalence relation between non-empty subsets $A , B$ of $\{ 1,2,3, 4,\dots,100\}$ defined by the condition:
the greatest element of $A$ is the same as the greatest element of $B .$ Let $P$ be the partition corresponding to this equivalence relation. What are the elements of $P$?
Can someone please explain this question to me? Does the first line means we can have any element in $ A$ and $B$ as long as the greatest of both are equal? and what is $P$ that corresponds to the equivalence relation? What is the equivalence relation?
Hint: Here is one equivalence class (which in turn is a set of the partition): $\{\{3\},\{1,3\},\{2,3\},\{1,2,3\}\}$. This set of the partition contains all subsets of $\{1,2,\dots,100\}$ with max element $3$.