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Chapter Three Relations 3.1. Relations. Relation is an important fundamental concept in the theory of sets. Let recall that if x , y R . Then x y means that y x is a nonnegative numbers. If we have an ordered pair (x , y ) of real numbers, then we can determine that x y , we check if y x is non-negative. Hence the x y describe a property of the order pair (x , y ) . We call it R of R 2 defined as follows: R (x , y ) R 2 x y (x , y ) R 2 y x is non-negative The set R determines the relationship between x and y relative to the property less than or equal. Hence x y if and only if (x , y ) R . Generalizing this we get the concept of relation. Definition 3.1. A relation R between a set A and a set B is a subset of A B , that is, R is a relation between A and B if and only if R A B . We simply write (a, b ) R or aRb for a A , and b B . R is not relation between A and B if and only if R A B , that is (a, b ) R or a R b if and only if R A B . A relation R on a set A is a subset of A 2 A A . We note that if A is a set with m elements and B is a set with n elements, then there are 2mn relations from A to B . Example 3.1. Let A 1, 2 and B 3, 4 . Then A B (1,3),(1, 4),(2,3),(2, 4) Then R (2,3),(2, 4 A B is a relation between the A and B . Hence (2,3) R (or 2R 3) and (2, 4) R (or 2R 4) . Example 3.2. Let A 1, 2 . Then A 2 (1,1),(1, 2),(2,1),(2, 2) . Since A 2 has 4 elements, (A 2 ) has 24 16 subsets, thus 16 relations on A 1, 2 . Here are several relations on A . R1 (1,1),(2, 2) R 2 (1,1),(1, 2) R 3 (1,1),(1, 2),(2, 2) 22 R 4 (1, 2) R 5 (2,1),(2, 2) Example 3.3. Let A 1, 2,3 and B a,b . Then R1 (1, a),(1, b ),(2, a),(2, b ) R 2 (3, a),(3, b ) R 3 (1, a),(1, b ),(3, a) are relations from A to B . It is clear that 1R1a, and 1R 3a , but 1 R 2 a . Definition 3.2. Let A be a set , and I A (a, a ) a A A 2 . Then I A is called the identity relation. Example 3.5. Let A 1, 2,3 . Then I A (1,1),(2, 2),(3,3) A 2 is the identity relation on A . Definition 3.3. Every relation R from a set A to a set B has an inverse relation R 1 from B to A which is defined by R 1 (b , a ) (a, b ) R , a A , b B Example 3.6. Let A 1, 2,3 and B a,b . Then R1 (1, a),(1, b ),(2, a),(2, b ) , and R11 (a,1),(b ,1),(a, 2), (b , 2) R 2 (3, a),(3, b ) , and R 21 (a,3),(b ,3) R3 (1, a),(1, b ),(3, a) , and R 31 (a,1),(b ,1),(a,3) Then R11 , R 21 and R 31 are inverse relations from B to A . 3.2. Properties of relations We continue our study of relations by looking to various properties of relations. Definition 3.4. A relation R on a set A is called reflexive if aRa , that is (a, a ) R for all a A . Hence every element is related to itself. We note that the identity relation I A (a, a ) a A A 2 is reflexive. Example 3.7. Let A 1, 2,3 . Then R1 (1,1),(1, 2),(2, 2),(2,3),(3,3) is a reflexive relation, but R 2 (1,1),(1, 2) is not a reflexive relation. 23 Example 3.8. Let R be the relation on the real numbers R defined by the sentence x y for all x , y R . Then R is not reflexive, for x x for any real number x . Example 3.9. Let be a family of sets, and let R be the relation on defined by the sentence A is a subset of B . Then R is a reflexive relation on , for every set is a subset of itself. Definition 3.5. A relation R on a set A is called symmetric if aRb implies that bRa , that is (a , b ) R implies (b , a ) R for all a , b A . Hence R is symmetric if a is related to b , then b is also related to a . We note that R is a symmetric relation on A if and only if R R 1 , for if (a, b ) R , then (b , a) R 1 . Example 3.10. Let A 1, 2,3, 4 . Then R1 (1,3),(2, 4),(3,1),(4, 2) is a symmetric relation, but R 2 (1,3),(2, 4),(3,1) is not a symmetric relation, for (2, 4) R 2 , but (4, 2) R 2 . Example 3.11. Let R be the relation on the natural numbers N which is defined by x divides y . Then R is not a symmetric relation, since 2 divides 4 , but 4 does not divides 2 , that is (2, 4) R but (4, 2) R . Definition 3.6. A relation R on a set A is called transitive if aRb and bRc implies that aRc , that is (a , b ) R and (b , c ) R implies (a, c ) R for all a, b , c A . Hence R is transitive if a is related to b , and b is related to c , then a is also related to c . Example 3.12. Let R be the relation on the real numbers R defined by the sentence x y for all x , y R . Then R is reflexive, for a b , and b c , then a c . Example 3.13. Let be a family of sets, and let R be the relation on defined by the sentence A is a subset of B . Then R is a transitive relation on , for if A B and B C , then A C . Example 3.14. Let A 1, 2,3, 4 . Then R1 (1,3),(3, 4),(1, 4) is a transitive relation, but R 2 (1,3),(3, 4) is not a transitive relation, for (1, 4) R 2 . 24 Definition 3.7. A relation R on a set A is called anti-symmetric if aRb and bRa imply a b , that is (a , b ) R and (b , a ) R imply a b , for all a , b A . Hence R is anti-symmetric if a is related to b , and b is also related to a , then a b . We note that if a b , then possibly a is related to b or possibly b is related to , but never both. Example 3.15. Let A 1, 2,3, 4 . Then R1 (1,1),(1,3),(3, 4),(4, 4) is an anti-symmetric relation, but R 2 (1,3),(3,1),(3, 4) is not anti-symmetric, since (1,3) R 2 and (3,1) R 2 , but 1 3 . Example 3.16. Let be a family of sets, and let R be the relation on defined by the sentence A is a subset of B . Then R is an antisymmetric relation on , for if A B and B A , then A B . Definition 3.8. A relation R on a set A is called irreflexive if a R a , that is (a, a ) R for a A . Hence every element is not related to itself. Definition 3.9. A relation R on a set A is called asymmetric if aRb imply b R a , that is (a , b ) R imply (b , a ) R , for all a , b A . Hence R is asymmetric if a is related to b imply b is not related to a . Example 3.17. Let A 1, 2,3 . Then the relation R1 (1,1),(1, 2),(2,1),(2,3) is neither reflexive, irreflexive, symmetric, asymmetric, symmetric, nor transitive. Let consider the relation anti- R 2 (1,1),(2, 2),(1, 2),(3,3) Then R 2 is reflexive, anti-symmetric, and transitive, but not symmetric. Definition 3.10. A relation R on a set A is called an equivalence relation if 1. R is reflexive, that is aRa for all a A . 2. R is symmetric, that is if aRb implies bRa ,for all a , b A . 3. R is transitive, that is if aRb and bRc , then aRc for all a, b , c A . If R is an equivalence relation on a set A , and aRb , then we say that a is equivalent to b . Example 3.18. An important example of an equivalence relation is that of equality, for any elements in any set, we have 1. a a 25 2. a b b a . 3. a b and b c a c . Example 3.19. Let A 1, 2,3, 4,5,6 and consider the relation R (1,1),(2, 2),(3,3),(4, 4),(5,5),(6,6),(2,3),(3, 2) Then R is an equivalence relation on A . Example 3.20. Let n be a fixed positive integer in Z . Define the relation n on Z by for all x , y Z , [x n y ] [n (x y )] [x n y ] [x y kn , for some k Z] [x n y ] [x y kn , for some k Z] Sometimes, we write [x y (mod n )] [n (x y )] [x y (mod n )] [x y kn , for some k Z] [x y (mod n )] [x y kn , for some k Z] Now, we show that n is an equivalence relation on Z . 1. For all x Z , we have x x 0 0n , for 0Z . Hence, for all x Z , we get x n x . Thus, n is a reflexive relation. 2. Let x , y Z , and suppose that x n y . Then n (x y ) , and hence there exist k Z such that x y kn . Thus y x (k )n , and so n ( y x ) , that is y n x . Hence, n is a symmetric relation. 3. Let x , y , z Z , and suppose that x n y and y n z . Then n (x y ) and n ( y z ) , and hence, there exist k , l Z such that x y kn , and y z ln .Thus x z (k l )n , and so n (x z ) , for (k l ) Z . Hence, n is a transitive relation. Therefore, n is an equivalence relation on Z . Definition 3.11. Let R be an equivalence relation on a set A , and let a A . Then the set a a b A bRa is called the equivalence class determine by a with respect to R . The set of all equivalence classes is called A modulo R and denoted by A . R 26 Definition 3.12. Let A be a set and P be a collection of nonempty subsets of A . Then P is called a partition of A if the following properties satisfied: 1. for all X ,Y P , either X Y or X Y . 2. A X . X P Example 3.21. Let A 1, 2,3, 4,5,6 . Let X 1 , Y 2, 4,6 , and X Y Z , and A X Y Z . So Z 3,5 . Then, P X ,Y , Z . Theorem 3.1. Let R be an equivalence relation on the set . Then 1. for all a A , a , 2. if b a , then a b , where, a , b A , 3. for all a , b A , either a b or a b , 4. A a , that is, A is the union of all equivalence classes with aA respect to R . 5. The set P a a A is a partition of A . Proof. 1. Let a A . Since R is reflexive, we have aRa Hence, a a [a ] and so a . 2. Let b A such that b a . Then bRa and by symmetric property of R , we have aRb . Now, if x a , then xRa . But aRb and R is transitive for R is an equivalence relation. Then xRb , and hence x b . Therefore, a b . Similarly, x b , then xRb . But bRa and R is transitive for R is an equivalence relation. Then xRa , and hence x a . Therefore, b a . Thus a b for all a , b R . 3. Let a , b R and a b . Then there exist x a b and hence, x a and x b . Thus xRa and xRb . Since R is symmetric, we have bRx , and by transitivity, we have bRa , and b a . Hence, by (2), a b . aA. 4. Let a A . Then a a a . Thus A a . Also, aA Hence, A aA aA a . aA Example 3.22. Let n be a fixed positive integer in Z . Define the relation n on Z by for all x , y Z , 27 [x n y ] [n (x y )] As in Example 3.20. Let Zn a a Z . By Theorem 3.1, we have Z n is a partition of Z . Then Zn 0,1, 2,..., n 1 , where, 0 0 kn kn k Z 1 1 kn k Z 2 2 kn k Z n 1 (n 1) kn k Z Suppose that n 5 . Then 0 5k k Z ..., 10, 5, 0,5,10,... 10 5 5 10 1 1 6k k Z ..., 9, 4,1, 6,11,... 9 4 6 11 2 2 kn k Z ..., 8, 3, 2, 7,12,... 8 3 7 12 3 3 kn k Z ..., 7, 2,3,8,13,... 7 2 8 13 4 3 kn k Z ..., 6, 1, 4,9,14,... 6 1 9 14 Hence, Z5 0,1, 2,3, 4 .Similarly, we find that Z8 0,1, 2,3, 4,5, 6, 7 . Let The A 1, 2,3, 4 . R (1,1),(2, 2),(3,3)(4, 4),(4, 2),(2, 4) . Then R is an equivalence relation on A . The distinct equivalence relations are Example 3.23. a [a ] x A (a, x ) R 1 [1] x A (1, x ) R 1 2 [2] x A (2, x ) R 2, 4 4 3 [3] x A (3, x ) R 3 Example 3.24. Let A 0,1, 2,3, 4 .Then R (0,0),(1,1),(2, 2),(3,3),(4, 4),(0, 4),(1,3),(4,0),(3,1) is an equivalence relation, and the distinct equivalence classes are: a [a ] x A (a, x ) R 0 [0] x A (1, x ) R 0, 4 4 1 [1] x A (1, x ) R 1,3 3 2 [2] x A (2, x ) R 2 Example 3.25. Consider the set of all real numbers , and the relation R (x , y ) R R x 2 y 2 0 1. We want to show that R is reflexive. Let x R . Then 0 x 2 x 2 . Thus (x , x ) R , that is xRx . Therefore, R is reflexive. 28 2. Now, we show that R is symmetric. Let x , y R such that (x , y ) R . Then x 2 y 2 0 , and this implies that y 2 x 2 0 . Hence ( y , x ) R , that is yRx . Therefore, R is symmetric. 3. Finally, we show that R is transitive. Let x , y , z R such that (x , y ) R and ( y , z ) R . Thus x 2 y 2 0 and y 2 z 2 0 .So x 2 y 2 y 2 z 2 0 , and x 2 z 2 0 . Thus (x , z ) R , that is xRz . Therefore, R is transitive. Therefore, R is reflexive, symmetric and transitive, and hence, R is an equivalence relation. The equivalence classes has the form x [x ] y R yRx y R ( y , x ) R y R y y R y 2 x 2 0 2 x2 y R y x x , x For any x R . If x 4 , then 4 [4] y R ( y , 4) R y R y 4 4, 4 Similarly, 3 [3] 3,3 . Example 3.26. Let A be any set. Then the identity relation I A (a , a ) a A A 2 is an equivalence relation and the equivalence classes are a [a ] b A a I A b b A (a , b ) I A b A a b a for any a A . If a , b A such that a b , then a b , that is a b and the two classes are distinct. If A 1, 2,3, 4 , then the equivalence classes for the identity relation I A are 1 , 2 , 3 , and 4 . 3.3. Domain and range of relations Let R be a relation between a set A and a set B . Then R A B . The domain of D R of the relation R is the set D R a A (a, b ) R , b B A 29 The range Rang R of the relation R is the set Rang R b B (a , b ) R , a A B Example 3.27. Let A 1, 2,3, 4 , and B 5,6,7 . Consider the relation R (2,5),(4,5),(4,7) between the set A and the set B . Then the domain of R is DR 2, 4 and the range of R is Rang R 5,7 . Also, R 1 (5, 2),(5, 4),(7, 4) is a relation between B and A . domain of R 1 is D R 1 5,7 , and the range of R Example 3.28. Let consider the relation (x , y ) x , y R,9 y 1 The is Rang R 1 2, 4 . R (x , y ) x , y R, 4x 2 9 y 2 36 R R 2 36 4x 2 4 ( x , y ) x , y R, y 2 4 x 2 9 4 ( x , y ) x , y R, y 4 x 2 9 4 ( x , y ) x , y R, 4 x 2 0 9 4 ( x , y ) x , y R, 4 x 2 9 ( x , y ) x , y R, x 2 9 (x , y ) x , y R, x [ 3,3] Thus D R [3,3] . Similarly, (x , y ) x , y R, 4x R (x , y ) x , y R, 4x 2 9 y 2 36 R R 2 36 9 y 2 9 ( x , y ) x , y R, x 2 9 y 2 4 9 ( x , y ) x , y R, x 9 y 2 4 9 (x , y ) x , y R,9 y 2 0 4 9 (x , y ) x , y R,9 y 2 4 ( x , y ) x , y R, y 2 4 (x , y ) x , y R, y [ 2, 2] 30 Thus, Rang R [2, 2] . Furthermore, R 1 (x , y ) x , y R,9x 2 4 y 2 36 Hence D R 1 [2, 2] , and Rang R 1 [3,3] . Example 3.29. Let R be the relation defined on the natural numbers N such that R (x , y ) x , y N,3x y 15 (x , y ) x , y N, y 15 3x (1,12), (2,9), (3, 6), (4,3) R 1 (12,1),(9, 2),(6,3),(3, 4) Note that x 5, y 0 is a solution for 3x y 15 in general, but (5,3) R for 0 N.. Hence, DR 1, 2,3, 4 Rang R 1 , and Rang R 12,9,6,3 DR 1 . 3.4. Partially and totally ordered sets Definition 3.13. Let R be a relation on a set A . Then R is called a partial order relation, if it satisfies the following conditions: 1. R is reflexive, that is aRa for all a A . 2. R is anti-symmetric, that is if aRb and bRa , then a b for all a,b A . 3. R is transitive, that is if aRb and bRc , then aRc for all a, b , c A . If a relation R on a set A defines a partial order on A , then aRb is denoted by a b which reads a precedes b , and we write (A , R ) (A , ) to denote the set A and the partial ordering R , and we call A is a partially ordered set (poset). Example 3.30. Let consider the set R . Define the relation on R , R (x , y ) x , y R, x y . We know that is reflexive, anti-symmetric, and transitive. So R is a partial order relation on R , that is (A , R ) . Example 3.31. Let be a family of sets. Define the relation on the set , R (A , B ) A , B , A B . We know that is reflexive, anti-symmetric, and transitive. Thus R is a partial order relation on , that is (A , R ) . Example 3.32. Let consider the set of all real numbers R . Define the relation on R , R (x , y ) x , y R, x y . We know that < is not reflexive. Thus R is not a partial order relation on R . 31 Example 3.33. Let A 1, 2,3 , and the relation R (1,1),(2, 2),(3,3),(1, 2 on A . Thus R is reflexive, anti-symmetric, and transitive, that is R is a partial order relation on A . Example 3.34. Let A l l is a line in the plane . Define R (l1 , l 2 ) l1 , l 2 A , l1 l 2 , that is l1 is perpendicular to l 2 A A Then R is not a partial order on A , since any line can not be a perpendicular to itself, that is (l , l ) R , and hence R is not reflexive. Definition 3.14. (i) Let R be a partially ordered relation on a set A . Elements a , b A are said to be comparable if and only if either aRb or bRa . Otherwise, a and b are non-comparable. (ii) A partially ordered relation R on a set A is called a totally order relation, if either aRb or bRa for any a , b A . Example 3.35. A partially ordered relation on any set A of real numbers is totally ordered set. a Example 3.36. Let A 1, 2,3, 4,5,6 . Define the relation R (x , y ) x , y A , x divieds y (x , y ) x , y A , x y (1,1), (1, 2), (1,3), (1, 4), (1,5), (1, 6), (2, 2), (2, 4), (2, 6), (3,3), (3, 6), (4, 4), (5,5), (6, 6) Then R is a partial order relation , but not a total order, since (3,5) R and (5,3) R . Solved Problems 1. Let A 1, 2,3, 4 , and B 1,3,5 . Define a relation R between A and B by R (x , y ) x A , y B ; x y . Write the relation R . Solution. R (x , y ) x A , y B ; x y (1,3), (1,5), (2,3), (2,5), (3,5) . 2. Let A 2,3, 4,5 , and B 3,6,7,10 . Define a relation R between A and B by R (x , y ) x A , y B ; x y . Write the relation R , R 1 ,and find the domain and range R . R (x , y ) x A , y B ; x y (2, 6), (2,10), (3,3), (3, 6), (5,10) . Solution. Also R 1 (6, 2),(10, 2),((3,3),(6,3),(10,5) . Then DR 2,3,5 Rang R 1 and Rang R 3,6,10 D R 1 . 3. Answer the following True (T) , or false (F): 32