1. 1 Discrete Structures – CNS2300 Text Discrete Mathematics and Its Applications (5 th Edition) Kenneth H. Rosen Chapter 7 Relations

2. 2 Section 7.5 Equivalence Relations

3. 3 A relation on a set A is called an equivalence relation if it is reflexive, symmetric, and transitive.

4. 4 I WISH THIS SLED HAD A SPEEDOMETER SO WE COULD KNOW HOW FAST WE’RE GOING.

5. 5 I SUPPOSE WE COULD MEASURE THE HILL. TIME OUR DESCENT, CALCULATE OUR RATE IN FEET PER MINUTE, AND CONVERT THAT INTO MILES PER HOUR.

6. 6 UM, YES.. THAT SOUNDS LIKE MATH.

7. 7 SUDDENLY I STOPPED CARING.

8. 8 Examples: Let R be the relation on the set of strings such that a R b if and only if l (a) = l (b) where l (x) is the length of x. Let R be the relation on the set of real numbers such that a R b if and only if a-b is an integer.

9. 9

10. 10 Equivalence Classes Let R be an equivalence relation on a set A. The set of all elements that are related to an element a of A is called the equivalence class of a. The equivalence class of a with respect to R is denoted by [a] R. When only one relation is under consideration, we will delete the subscript R and write [a] for this equivalence class.

11. 11 [0] = {…-14, -7, 0, 7, 14, 21,…} [1] = {…-13, -6, 1, 8, 15, 22,…} [2] = {…-12, -5, 2, 9, 16, 23,…} [3] = {…-11, -4, 3, 10, 17, 24,…} [4] = {…-10, -3, 4, 11, 18, 25,…} [5] = {…-9, -2, 5, 12, 19, 26,…} [6] = {…-8, -1, 6, 13, 20, 27,…}

12. 12 [0] 7 = {…-14, -7, 0, 7, 14, 21,…} [1] 7 = {…-13, -6, 1, 8, 15, 22,…} [2] 7 = {…-12, -5, 2, 9, 16, 23,…} [3] 7 = {…-11, -4, 3, 10, 17, 24,…} [4] 7 = {…-10, -3, 4, 11, 18, 25,…} [5] 7 = {…-9, -2, 5, 12, 19, 26,…} [6] 7 = {…-8, -1, 6, 13, 20, 27,…} Congruence Classes modulo m. [a] m

13. 13 Equivalence Classes and Partitions Let R be an equivalence relation on a set A. The following statements are equivalent:

14. 14 Partition A partition of a set S is a collection of disjoint, nonempty subsets of S that have S as their union.

15. 15 Partition A1A1 A2A2 A3A3 A4A4 A5A5 A6A6 A7A7 A8A8

16. 16 Let R be an equivalence relation on a set S. Then the equivalence classes of R form a partition of S. Conversely, given a partition of the set S, there is an equivalence relation R that has the sets as its equivalence classes.

17. 17 finished