Interactive Real Analysis

• Real Analysis
• 1. Sets and Relations
• 1.1. Notation and Set Theory
• 1.2. Relations and Functions
• 1.3. Equivalence Relations and Classes
• 1.4. Natural Numbers, Integers, and Rational Numbers
• 2. Infinity and Induction
• 3. Sequences of Numbers
• 4. Series of Numbers
• 5. Topology
• 6. Limits, Continuity, and Differentiation
• 7. The Integral
• 8. Sequences of Functions
• 9. Historical Tidbits
• Java Tools

1.3. Equivalence Relations and Classes

Example 1.3.4(b):

Consider the set Z of all integers. Define a relation r by saying that x and y are related if their difference y - x is divisible by m. Then we have:

Back

• This relation is an equivalence relation (i.e. the three conditions are satisfied)
• There are m equivalence classes:
  1. all numbers divisible by m with no remainder are in class 0
  2. all numbers divisible by m with remainder 1 are in class 1
  3. all numbers divisible by m with remainder 2 are in class 2
  4. ...
  5. all numbers divisible by m with remainder m-1 are in class m-1
• Addition can be defined by adding modulo m. That is, if we consider the equivalence classes obtained by dividing the differences by, say, 5, then we have, as an example:
  • [(2)] + [(1)] = [(3)]
  • [(2)] + [(4)] = [(1)]
  • [(3)] + [(4)] = [(3)]
  • etc...
• The (important) details are left as an exercise.