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Exploring Equivalence Classes in Mathematics

Learn about equivalence relations, classes, and partitions in mathematics. Explore definitions, examples, theorems, and applications. Understand how equivalence relations partition a set into equivalent subsets. Discover the concept of equivalence classes and their representatives.

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Exploring Equivalence Classes in Mathematics

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  1. Equivalence Relations MSU CSE 260

  2. Outline • Introduction • Equivalence Relations • Definition, Examples • Equivalence Classes • Definition • Equivalence Classes and Partitions • Theorems • Example

  3. Introduction • Consider the relation R on the set of MSU students: aRb a and b are in the same graduating class. • R is reflexive, symmetric and transitive. • Relations which are reflexive, symmetric and transitive on a set S, are of special interest because they partition the set S into disjoint subsets, within each of which, all elements are all related to each other (or equivalent.)

  4. Equivalence Relations • Definition.A relation on a set A is called an equivalence relation if it is reflexive, symmetric, and transitive. • Two elements related by an equivalence relation are called equivalent.

  5. Example • Consider the Congruence modulo m relation R = {(a, b)  Z | a  b (mod m)}. • Reflexive. a  Za R a since a - a = 0 = 0  m • Symmetric. a, b  Z If a R b then a - b = km. So b - a = (-k) m. Therefore, bRa. • Transitive. a, b, c  Z If a R b  b R c then a - b = km and b - c = lm. So (a-b)+(b-c) = a-c = (k+l)m. So a R c. • R is then an equivalence relation.

  6. Equivalence Classes • Definition.Let R be an equivalence relation on a set A. The set of all elements related to an element a of A is called the equivalence class of a, and is denoted by [a]R. • [a]R = {xA | (a, x)  R} • Elements of an equivalence class are called its representatives.

  7. Example • What are the equivalence classes of 0, 1, 2, 3… for congruence modulo 4? • [0]4 = {…, -8, -4, 0, 4, 8, …} • [1]4 = {…, -7, -3, 1, 5, 9, …} • [2]4 = {…, -6, -2, 2, 6, 10, …} • [3]4 = {…, -5, -1, 3, 7, 11, …} The other equivalence classes are identical to one of the above. [a]m is called the congruence class of a modulom.

  8. Equivalence Classes & Partitions • Theorem. Let R be an equivalence relation on a set S. The following statements are logically equivalent: • a R b • [a] = [b] • [a][b]  

  9. Equiv. Classes & Partitions - cont • Definition. A partition of a set S is a collection {Ai | i  I} of pairwise disjoint nonempty subsets that have S as their union. • i,jIAi  Aj = , and iIAi= S. • Theorem. Let R be an equivalence relation on a set S. Then the equivalence classes of R form a partition of S. Conversely, for any partition {Ai | i  I} of S there is an equivalence relation that has the sets Ai as its equivalence classes.

  10. Example • Every integer belongs to exactly one of the four equivalence classes of congruence modulo 4: • [0]4 = {…, -8, -4, 0, 4, 8, …} • [1]4 = {…, -7, -3, 1, 5, 9, …} • [2]4 = {…, -6, -2, 2, 6, 10, …} • [3]4 = {…, -5, -1, 3, 7, 11, …} • Those equivalence classes form a partition of Z. • [0]4  [1]4  [2]4  [3]4 = Z • [0]4, [1]4,[2]4 and[3]4 are pairwise disjoint.

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