Congruence as an equivalence relation

• Read carefully pages 202 through 207 (section 8.5)
• Some key questions to answer:
  1. When do we say that two numbers \(a\), \(b\) are congruent modulo \(n\)?
  2. Show that for any natural number \(n\geq 2\), congruence modulo \(n\) defines an equivalence relation on \(\mathbb{R}\). This is a really important proof.
  3. What happens when \(n=1\)? Describe what congruence modulo \(1\) would mean.
  4. Explain in simple terms what congruence modulo \(2\) means. We have already worked with the more straightforward explanation of it.
  5. Define a relation \(R\) on \(\mathbb{Z}\) by saying that \(aRb\) if and only if \(2a+b\) is congruent to \(0\) modulo \(3\). Show that this is an equivalence relation.
• Practice problems from section 8.5 (page 213): 8.45, 8.46, 8.49