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MAT 2720 Discrete Mathematics Section 3.3 Relations http://myhome.spu.edu/lauw Goals Relations • Properties of Relations on X Recall A relation from X to Y is a subset R X Y Sometimes, we write xRy if x, y R Domain of R = all possible value of x Range of R = all possible value of y Recall A relation from X to X is called a relation on X R XX Properties of Relation on X R is… Reflexive Symmetric Transitive If… Diagraph Example 5(a) X {1, 2,3, 4}; R X X R 1,1 , 1,3 , 2, 2 , 3, 4 Reflexive? Example 5(b) X {1, 2,3, 4}; R X X R 1,1 , 1,3 , 2, 2 , 3, 4 Symmetric? Example 5(c) X {1, 2,3, 4}; R X X R 1,1 , 1,3 , 2, 2 , 3, 4 Transitive? Properties of Relation on X R is… Antisymmetric If… Diagraph (Read) MAT 2720 Discrete Mathematics Section 3.4 Equivalence Relations http://myhome.spu.edu/lauw Goals Equivalence Relations • A special relation with nice properties. • Partition of sets (Clumping Property). • Applications to counting problems. CS students should read the applications in p.166-168 “Informal” Example X The 30 days of April of 2008 , R XX ( x, y ) R if x and y are on the same "position" of the week. Example ( , ) R ( , ) R ( , ) R “Informal” Example X The 30 days of April of 2008 , R XX ( x, y ) R if x and y are on the same "position" of the week. Reflexive? “Informal” Example X The 30 days of April of 2008 , R XX ( x, y ) R if x and y are on the same "position" of the week. Symmetric? “Informal” Example X The 30 days of April of 2008 , R XX ( x, y ) R if x and y are on the same "position" of the week. Transitive? “Informal” Example X The 30 days of April of 2008 , R XX ( x, y ) R if x and y are on the same "position" of the week. “Clumping” Effective Definitions and Notations is an Equivalence Relation if R is reflexive, symmetric, and transitive. R XX Example 1 X 1, 2,3, 4,5, 6, 7,8,9,10 , R X X x, y R if 3 divides x y Show that R is an Equivalence Relation Example 1 X 1, 2,3, 4,5, 6, 7,8,9,10 , R X X x, y R if 3 divides x y Show that R is an Equivalence Relation 3 divides x y Example 1 X 1, 2,3, 4,5, 6, 7,8,9,10 , R X X x, y R if 3 divides x y Proof: Reflexive Analysis Example 1 X 1, 2,3, 4,5, 6, 7,8,9,10 , R X X x, y R if 3 divides x y Proof: Symmetric Analysis Example 1 X 1, 2,3, 4,5, 6, 7,8,9,10 , R X X x, y R if 3 divides x y Proof: Transitive Analysis Definitions and Notations is an Equivalence Relation if R is reflexive, symmetric, and transitive. Equivalence Class of a X : a x | xRa x | x, a R R XX Example 1 X 1, 2,3, 4,5, 6, 7,8,9,10 , R X X x, y R if 3 divides x y [1] [2] [3] [1] [2] [3] Observations X 1, 2,3, 4,5, 6, 7,8,9,10 , R X X x, y R if 3 divides x y [1] [2] [3] 1. 2. [1] [2] [3] Observations X 1, 2,3, 4,5, 6, 7,8,9,10 , R X X x, y R if 3 divides x y [1] [2] [3] 1. 2. [1] [2] [3] 1 4 7 10 2 8 5 6 3 9 X Partition of a Set (1.1) A partition of a set X is a way to split X into the union of disjoint subsets. S A, B, C, D, E B A E C D X Partition of a Set (1.1) A partition of a set X is a way to split X into the union of disjoint subsets. For every element in X, S A, B, C, D, E it belongs to one and only one subset in the partition. B A E C D X Theorem Let R X X be an equivalence relation. Then S [a] | a X forms a partition on X . b a a b c c X “Informal” Example X The 30 days of April of 2008 , R XX ( x, y ) R if x and y are on the same "position" of the week. Partition 1 1,8,15,22,29 5 5,12,19,26 2 2,9,16,23,30 6 6,13,20,27 3 3,10,17,24 7 7,14,21,28 4 4,11,18,25 Theorem Let S be a partition on X , define R X X by x, y R if x, y belong to the same subset in S , then R is an equivalence relation. S A, B, C, D, E B y x A E C D X Example 2 X 1, 2,3, 4,5, 6 , A 1 , B 2, 4 , C 3,5, 6 1 2 3 S { A, B, C} 4 5 6 R (It is easy to check that R is an equivalence relation.) Summary of the 2 Theorems Partition Equivalence Relation Theorem Let R be an equivalence relation on a finite set X , If each equivalence class has r elements, then there are X r equivalence class Theorem Let R be an equivalence relation on a finite set X , If each equivalence class has r elements, then there are X r equivalence class X A1 A2 A3 Ak