1. Relations — on a nonempty set

2. Relations Definition: A relation on a nonempty set A is a nonempty set R of ordered pairs (x, y), where x, y  A. If (a, b)  R, we write aRb.

3. Relations Definition: A relation R on a set A is an equivalence relation if the following properties are satisfied: For x, y, z  A 1. Reflexive: xRx  x  A [(x, x)  R.] 2. Symmetric: If xRy, then yRx, [If (x, y)  R then (y, x)  R.] 3. Transitive: If xRy and yRz, then xRz,. [If (x, y)  R and (y, z)  R then (y, x)  R.]

4. Relations Ex1. Let A = {  2,  5, 2, 5} and R = {(5,  2), (5, 2), (  5, 2), (  5,  2)}. Then 5R2,  5R2, 5R(  2), but 2R5, 5R5. The relation R is neither reflexive nor symmetric properties, but it satisfies the transitive propert.

5. Relations Ex2. (a) A 為目前學籍在義守大學的人所成的集合 xR 1 y if and only if x 和 y 屬於同一系級 xR 1 ’y if and only if x 和 y 本學期修同一門課

6. Relations Ex2. (b)&(c) A 為現存世界上的 (4 輪 ) 車子所成的集合 aR 2 b if and only if a 和 b 屬於同一廠牌 A 為現今全世界還活著的人所成的集合 aR 3 b if and only if a 和 b 有婚姻關係（不論 仍在繼續或已終止）

7. Relations Ex2. (d) Let A = Z, and xR 4 y if and only if  x  =  y . Then (1,  1)  R 4, (5, 5)  R 4 and (1, 3)  R 4. This is an equivalence relation. 1. reflexive: 2. symmetric: 3. transitive:

8. Relations Ex2. (e) Let A = Z, xR 5 y if and only if x > y. (1,  1)  R 5, (5, 5)  R 5 and (1, 3)  R 5. This is not an equivalence relation, since it does not satisfy reflexive and symmetric property.

9. Relations Ex2. (f) Let A = Z, xR 6 y if and only if x − y is an even number. (1,  1)  R 6, (5, 5)  R 6, (2, 5)  R 6. This is an equivalence relation. 1. reflexive: 2. symmetric: 3. transitive:

10. Relations Ex2. (g) Let m be a positive integer and A = Z. x R 7 y if and only if x − y is a multiple of m. Use the same argument as Ex2. (f), we can show that R 7 is an equivalence relation on Z.

11. Relations Definition: Let R be an equivalence relation on the nonempty set A. For each a  A, then set [a]={x  A ｜ xRa} is called the equivalence class of the relation R.

12. Relations Ex3. (a) A 為目前學籍在義守大學的人所成的 集合。 xR 1 y if and only if x 和 y 同一系級。 每一個系級所成的集合都為一個 R 1 中的 equivalence class 。 (b) A 為現在存在世界上的 (4 輪 ) 車子所成 的集合。 aR 2 b if and only if a 和 b 屬於同一 廠牌。則同一個廠牌的車子所成的集合 都是 R 2 中的 equivalence class 。

13. Relations Ex3. (c) Let A = Z, and xR 4 y if and only if  x  =  y . 每一個 equivalence class 都包含 兩個元素。例如： (d) Let A = Z, xR 6 y if and only if x − y is an even number. 在 A 中只有兩個 equivalence classes 。

14. Relations Ex3. There are m different equivalence classes of R 7 in A. Assume m = 4. Then the 4 different equivalence classes are: [0] = [1] = [2] = [3] = Z = [0]  [1]  [2]  [3] (disjoint union). We say that {[0], [1], [2], [3]} is a partition of Z.