Equivalence Classes

• Read carefully pages 198 through 202 (section 8.4)
• Some key questions to answer:
  0. True or False: An equivalence class \([a]\) for an element in an equivalence relation may be the empty set.
  1. Prove that if \(R\) is an equivalence relation on a nonempty set \(A\), and \(a\), \(b\) are two elements of \(A\), then the elements are related if and only if their equivalence classes are identical sets. This is a very important proof to understand.
  2. Prove that if \(R\) is an equivalence relation on a nonempty set \(A\), and \(a\), \(b\) are two elements of \(A\), then their equivalence classes are either identical sets (equal as sets) or are disjoint. Another way to phrase is that if \([a]\cap[b]\neq\emptyset\) then \([a] = [b]\).
  3. Prove that the equivalence classes of an equivalence relation on a set \(A\) form a partition of \(A\).
  4. Show that if we have a partition on a set \(A\), then we can use that partition to define an equivalence relation on \(A\) in such a way that the equivalence classes are exactly the subsets of \(A\) that are the elements of the partition. Many steps to this:
     • Define the relation \(R\).
     • Prove that it is an equivalence relation (reflexive, symmetric, transitive).
     • Show that if \(a\in B\) and \(B\) is one of the subsets in the partition, then \([a] = B\).
• Practice problems from section 8.4 (page 212): 8.36, 8.37, 8.39, 8.41, 8.42