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Relations —

Relations —. on a nonempty set. 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. Definition:. A relation R on a set A is an equivalence relation

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Relations —

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  1. Relations — on a nonempty set

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

  3. 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.] Relations

  4. 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. Relations

  5. Ex2. (a) • A為目前學籍在義守大學的人所成的集合 • xR1y if and only if x和y屬於同一系級 • xR1’y if and only if x和y本學期修同一門課 Relations

  6. Ex2. (b)&(c) • A為現存世界上的(4輪)車子所成的集合 aR2b 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 aR3b if and only if a和b有婚姻關係（不論仍在繼續或已終止） Relations

  7. Ex2. (d) • Let A = Z, and xR4y if and only if x=y. Then (1, 1)R4, (5, 5)R4 and (1, 3)R4. • This is an equivalence relation. 1. reflexive: 2. symmetric: 3. transitive: Relations

  8. Ex2. (e) • Let A = Z, xR5y if and only if x > y. • (1, 1)R5, (5, 5)R5 and (1, 3)R5. • This is not an equivalence relation, since it does not satisfy reflexive and symmetric property. Relations

  9. Ex2. (f) • Let A = Z, xR6y if and only if x − y is an even number. • (1, 1)R6, (5, 5)R6, (2, 5)R6. • This is an equivalence relation. 1. reflexive: 2. symmetric: 3. transitive: Relations

  10. Ex2. (g) • Let m be a positive integer and A = Z. • x R7y if and only if x − y is a multiple of m. • Use the same argument as Ex2. (f), we can show that R7 is an equivalence relation on Z. Relations

  11. 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 classof the relation R. Relations

  12. Ex3. • (a) A為目前學籍在義守大學的人所成的集合。xR1y 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R1中的equivalence class。 • (b) A為現在存在世界上的(4輪)車子所成的集合。aR2b 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R2中的equivalence class。 Relations

  13. Ex3. • (c) Let A = Z, and xR4y if and only if x=y.每一個equivalence class都包含兩個元素。例如： • (d) Let A = Z, xR6y if and only if x − y is an even number. 在A中只有兩個equivalence classes。 Relations

  14. Ex3. • There are m different equivalence classes of R7 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 partitionof Z. Relations

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