Relations & Functions

Transcription

1 Relations & Functions Discrete Mathematics (MA 2333) Faculty of Science Telkom Institute of Technology Bandung - Indonesia

2 Relations Let A and B be sets. A binary relation from A to B is subset of A B. Notation: R (A B). The notation used for a binary relation is a R b or (a, b) R, which means a is related to b oleh by relation R arbor (a, b) R means a is not related to b by relation R. Set A is called domain of R, and set B is called range of R

3 Relations Example A = {2,3,5}, B = {4,6,9,,5} A B = {(2,4), (2,6), (2,9), (2,), (2,5), (3,4), (3,6), (3,9), (3,), (3,5), (5,4), (5,6), (5,9), (5,), (5,5)} Relation R defined by : (a, b) R if a divides b R = {(2,4), (2,6), (2,), (3,6), (3,9), (3,5), (5,), (5,5)} This example show that R (A B) (2,4) R (3,) R

4 Relations Relations from a set A to itself are of special interest A relation on the set A is a relations A to A. A relation on set A is a subset of A A.

5 Relations Let A = {, 2, 3, 4} Relation R defined by : (a, b) R if a divides b, a,b A R = {(,), (,2), (,3), (,4), (2,2), (2,4), (3,3), (4,4)}

6 Representing Relations Ordered Pair Example A = {2,3,5}, B = {4,6,9,,5} Relation R defined by : (a, b) R if a divides b R = {(2,4), (2,6), (2,), (3,6), (3,9), (3,5), (5,), (5,5)}

7 Representing Relations Arrows Diagram A = {2,3,5}, B = {4,6,9,,5} Relation R defined by : (a, b) R if a divides b

8 Representing Relations Table The first column represents the domain, the second column represents range A = {2,3,5}, B = {4,6,9,,5} Relation R defined by : (a, b) R if a divides b A B

9 Representing Relations Matrix Let R be the relation that relate set A = {a, a2,, am} and B = {b, b2,, bn}. Relation R can be represented using matrix M = [mij] b b 2 b n mn m m n n m m m m m m m m m m a a a L M M M M L L M = R b a R b a m j i j i ij ), (, ), (, M =

10 Representing Relations Matrix (Cont) A = {2,3,5}, B = {4,6,9,,5} Relation R defined by : (a, b) R if a divides b M =

11 Representing Relations 4. Directed Graph Directed graph is defined to represent a relation on a set (not between two sets) Every element of the set is represented by a vertex and every ordered pair is represented by an edge (arc) Let A = {, 2, 3, 4} Relation R defined by : (a, b) R if a divides b, a,b A R = {(,), (,2), (,3), (,4), (2,2), (2,4), (3,3), (4,4)}

12 Relations Representations 4. Directed Graph (Cont) 2 4 3

13 Properties of Binary Relations Reflexive A relation R on a set A is called refexive if (a, a) R for every element a A. A relation on A is reflexive if every element of A is related to itself

14 Properties of Binary Relations Example. Let A = {, 2, 3, 4}, and relation R defined by : (a,b) R if a b and a,b A Hence, R = {(, ), (2, ), (2, 2), (3, ), (3, 2), (3, 3), (4, ), (4, 2), (4,3), (4,4)} It is seen that (,), (2,2), (3,3), (4,4) are element of R. Therefore R is reflexive Example. Let A = {, 2, 3, 4}, and relation R defined by : (a,b) R if a > b and a,b A Relasi R = {(2, ), (3, ), (3, 2), (4, ), (4, 2), (4,3)} is not reflexive because (,), (2,2), (3,3), (4,4) are not element of R Example. Relation divides on set of positive integers is reflexive, because every positive integers divide on itself, so that (a, a) R for every a A.

15 Properties of Binary Relations Reflexive relation has matrix which main diagonal elements is, or m ii =, for i =,2,..,n When a reflexive relation is represented using directed graph, there will always be loop in every vertex O

16 Properties of Binary Relations Transitive A relation R on set A is called transitive if (a, b) R and (b, c) R, then (a, c) R, which a, b, c A. Example. Let A = {, 2, 3, 4}, and relation R defined on A, a. R = {(2, ), (3, ), (3, 2), (4, ), (4, 2), (4, 3)} is transitive

17 Properties of Binary Relations See table below : (a,b) (b,c) (a,c) (3,2) (2,) (3,) (4,2) (2,) (4,) (4,3) (3,) (4,) (4,3) (3,2) (4,2)

18 Properties of Binary Relations R = {(, ), (2, 3), (2, 4), (4, 2) } is not transitive because (2, 4) and (4, 2) R, but (2, 2) R, (4, 2) and (2, 3) R, but (4, 3) R. Relation R = {(, ), (2, 2), (3, 3), (4, 4) } is transitif Relation R = {(, 2), (3, 4)} transitive because there is not (a, b) R and (b, c) R so that (a, c) R. Relation that contain only one element, R = {(4, 5)} alaways transitive

19 Properties of Binary Relations Example. Relation divides on set of positive integers is transitive. Suppose a divides b, and b divides c. Then there are exist positive integers m and n, so that b = ma dan c = nb. Therefore c = nma, then a divides c. So, relation divides is transitive

20 Properties of Binary Relations When transitive relation represented in the form of matrix, transitive relation does not have particular characteristics on its matrix Transitive property on directed graph is described by a condition : if there is an edge from a to b and b to c, there will be directed edge from a to c

21 Properties of Binary Relations Symmetric and anti-symmetric A Relation R on a set A is called symmetric if for every a, b A, if (a, b) R, then (b, a) R Relation R on a set A is not symmetric if (a, b) R while (b, a) R.

22 Properties of Binary Relations A Relation R on a set A is called anti-symmetric if for every a, b A, if (a, b) R, and (b, a) R, then a = b Notice that the term symmetric is not the antonym for the term antisymmetric

23 Properties of Binary Relations Example. Let A = {, 2, 3, 4}, and relation R below defined on set A, then R = {(, ), (, 2), (2, ), (2, 2), (2, 4), (4, 2), (4, 4) } is symmetric because (, 2) and (2, ) R, (2, 4) and (4, 2) R. R is not anti-symmetric R = {(, ), (2, 3), (2, 4), (4, 2) } is not symmetric because (2, 3) R, but (3, 2) R. R is not anti symmetric R = {(, ), (2, 2), (3, 3) } is antisymmetric because = dan (, ) R, 2 = 2 and (2, 2) R, and 3 = 3 and (3, 3) R. R is symmetric.

24 Properties of Binary Relations Example. Relation divides on set of positive integers is not symmetric because if a devides b, b is not divides a, except if a = b. For example, 2 divides 4, but 4 is not devides 2. Therefore (2, 4) R but (4, 2) R. Example. Relation divides anty-symmetric because if a divides b and b divides a, then a = b. For example, 4 divides 4. therefore, (4, 4) R and 4 = 4.

25 Properties of Binary Relations Symmetric relation has matrix which elements under the main diagonal are the reflection of the elements above the main diagonal, or m ij = m ji =, for i =, 2,, n : Symmetric relation when represented in the by directed graph, has characteristics of : if there is an edge from a to b, there will be an edge from b to a

26 Properties of Binary Relations Anti-symmetric relation has matrix which elements characteristic is : if m ij = with i j, then m ji =. In other words, the matrix of the anti-symmetric relation meets the condition of : is one of m ij = or m ji = if i j Characteristic of a directed graph of an anti-symmetric relation is that there will never be two edges with different directions between two different vertices

27 Inverse of Relation Let R is a relation from set A to set B. Inverse of relation R, denoted by R, is a relation from B to A which defined by R = {(b, a) (a, b) R }

28 Inverse of Relation Example. Let P = {2, 3, 4} and Q = {2, 4, 8, 9, 5}. If we define relation R from P to Q by : (p, q) R if p divides q, so we obtain R = {(2, 2), (2, 4), (4, 4), (2, 8), (4, 8), (3, 9), (3, 5) } R is relation from Q to P with (q, p) R if q is p then we obtain R = {(2, 2), (4, 2), (4, 4), (8, 2), (8, 4), (9, 3), (5, 3) }

29 Inverse of Relation If M is a matrix representing a relation R then matrix which representing relation R, is obtained by finding transpose of matrix M M = N = M T =

30 Combining Relations Sinc relations from A to B are subsets of A x B, two relations from A to B can be combined any way two sets can be combined If R and R 2 are relations from A to B, then R R 2, R R 2, R R 2, and R R 2 are also relation from A to B.

31 Combining Relations Example. Let A = {, 2, 3} and B = {, 2, 3, 4}. Relation R = {(, 2), (3, 3), (4, 4)} Relation R 2 = {(, ), (, 2), (, 3), (, 4)} R R 2 = {(, )} R R 2 = {(, ), (2, 2), (3, 3), (, 2), (, 3), (, 4)} R R 2 = {(2, 2), (3, 3)} R 2 R = {(, 2), (, 3), (, 4)} R R 2 = {(2, 2), (3, 3), (, 2), (, 3), (, 4)}

32 Combining Relations Suppose that the relations Rand R2 on set A are represented by the following matrices R = R2 = M R R2 = MR MR2 = MR R2 = MR MR2 =

33 Composition of Relations Let R be the relation from set A to set B, and let S be relation from set B to set C. The composition of R and S, is denoted by S ο R, is relation from A to C which is defined by: S ο R = {(a, c) a A, c C, for a,b B, (a, b) R and (b, c) S}

34 Composition of Relations Example. Let R = {(, 2), (, 6), (2, 4), (3, 4), (3, 6), (3, 8)} is relation from set {, 2, 3} to set {2, 4, 6, 8} and S = {(2, u), (4, s), (4, t), (6, t), (8, u)} is relation from set {2, 4, 6, 8} to set {s, t, u}. Then composition of relation R and S is S ο R = {(, u), (, t), (2, s), (2, t), (3, s), (3, t), (3, u)} s t u

35 Composition of Relations If relation R and R2 are represented by matrices M R dan M R2, then matrix representing the composition of the two relations are M R2 ο R = M R M R2 It is done by changing product with (operator AND) and by changing the addition with (operator OR)

36 Composition of Relations Example. Let relations R and R2 defined on set A are represented by matrices R = R2 = The matrix representing R2 ο R is MR2 ο R = MR. MR2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

37 Equivalence Relations A relation on a set A is called an equivalence relation if it is reflexive, symmetric and transitive. Two elements that are related by an equivalence relation are called equivalent Example Suppose that R is the relation on the set of strings of English letters such that a R b if and only if l(a) = l(b), where l(x) is the length of the string x. Is R an equivalence relation? Solution : Since l(a) = l(b) it follows that ara whenever a is string, so that R is reflexive

38 Equivalence Relations Suppose that a R b, so that l(a)=l(b). Then b R a, since l(b) = l(a). Hence, R is symmetric Suppose that a R b and b R c. Then l(a)=l(b) and l(b)=l(c). Hence l(a)=l(c), so that arc Consequently R is an equivalence relation

39 Equivalence Relations Example Let R be the relation on the set of real number such thst arb if and only if a b is an integer. Is R an equivalence relation? Solution : Since a a = is an integer for all real numbers a, ara for all real numbers a. Hence, R is reflexive Suppose that a R b. Then a b is an integer, so that b a is also an integer. Hence, b R a. It follows that R is symmetric If a R b and b R c, then a b and b c are integers. Therefore, a-c = (a-b)+(b-c) is also integer. Hence a R c. Thus R transitive. So, R is an equivalence relation

40 Equivalence Relations Let A be the set of all students in IT Telkom. Consider the relation R on A that consists of all pairs (x,y) where x and y graduated from the same high school. Given a student x, we can form the set the set of all students equivalent to x with respect to R. This set contains of all students who graduated from the same high school as x did. This subset of A is called an equivalence class of the relation

41 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 [a] R = {s (a,s) R} When only one relation is under consideration, we will delete the subscript R and write [a] for this equivalence class If b [a] R, then b is called a representative of theis equivalence class

42 Partial Orderings A relation on a set S is called a partial ordering or partial order if it is reflexive, antisymmetric, and transitive A set S together with partial ordering R is called a partially ordered set (poset) and is denoted by (S,R) Example Show that the greater than or equal relation ( ) is aprtial ordering on the set of integers. Solution: We know that R = ( a, b) For every integers a, { a b} a it means, for every integers a, (a,a) R. thus R reflexive a

43 Partial Orderings For arbitrary integer a, b, if a b and b a, then a = b. It means if (a,b) R and (b,a) R, then a = b. Thus R is antisymmetric For arbitrary integer a, b, c, if a b and b c, then a c. It means If (a,b) R and (b,c) R then (b,c) R. Thus R is transitive. Hence, R is partial ordering, and (Z, ) is poset Exercise : Show that divisibility relation (denoted by ) is partial ordering on the set of positive integers Show that the inclusion relation is a partial ordering on the power set of a set S

44 Partial Orderings The element a and b of a poset (S,R) are called comparable if either a R b or b R a. When a and b are elements of S such that neither a R b nor b R a, a and b are called incomparable Example In the poset (Z +, ), are the integers 3 and 9 comparable? Are 5 and 7 are comparable? The integers 3 and 9 are comparable, since 3 divides 9 (3 9) The integers 5 and 7 are incomparable, because 5 does not divide 7 and 7 does not divide 5

45 Partial Orderings The adjective partial is used to describe partial orderings since pairs of elements may be incomparable. When every two elements in the set are comparable, the relation is called a total ordering If (S,R) is a poset and every two elements of S are comparable, S is called a totally ordered set, and R is called a total order

46 Partial Orderings The poset (Z, ) is totally ordered set, since a b or a b whenever a nd b integers The poset (Z +, ) is not totally ordered set since it contains elements that are incomparable

47 Hasse Diagram Hasse Diagram : Diagram which contains sufficient information to find the partial ordering Procedure to construct the Hasse Diagram. Start with directed graph for this relation. All edges are pointed upward 2. Because a partial ordering is reflexive, remove the loops 3. Remove all edges that that must be present because the transitivity 4. Remove all the arrows on the directed edges

48 Hasse Diagram Example Draw Hasse diagram representing partial ordering on set {, 2, 3, 4} R ( a b) { a b} =,. Start with directed graph for this relation. All edges are pointed upward 4 3 2

49 Hasse Diagram 2. Because a partial ordering is reflexive, remove the loops 4 3 2

50 Hasse Diagram 3. Remove all edges that that must be present because the transitivity 4 3 2

51 Hasse Diagram 4. Remove all the arrows on the directed edges. The result is Hasse diagram 4 3 2

52 Hasse Diagram Example Draw Hasse diagram representing partial ordering a, b a divides b on set {, 2, 3, 4, 6, 8, 2} ( ) { } Directed Graph Hasse Diagram

53 Functions

54 Definition Let A and B be sets Binary relation f from A to B is a function if every element in A related to exactly one element in B If f is a function from A to B, we can denote f : A B It means f assigns set A to B. A is called the domain of f and B is called codomain of f. Another term for function is mapping or transformation.

55 Definition If f(a) = b, then b is called image of a and a is called pre-image of b. The set that contains all values of mapping f is called range of f. A B f a b

56 Example Relation f = {(, u), (2, v), (3, w)} from A = {, 2, 3} to B = {u, v, w} is function from A to B. f() = u, f(2) = v, and f(3) = w. Domain of f is A and codomain is B. Range of f is {u, v, w}

57 Example Relation f = {(, u), (2, u), (3, v)} from A = {, 2, 3} to B = {u, v, w} is a function from A to B. Domain of f is A, codomain B, Range of f is {u, v}.

58 Example Relation f = {(, u), (2, v), (3, w)} from A = {, 2, 3, 4} to B = {u, v, w} is not function, because not all elements of A mapped to B.

59 Example Suppose f : Z Z defined by f(x) = x 2. Domain and codomain of f is set of integer, and range of f is a set of positive integer and zero

60 Function One to One Function f is called one-to-one or injective, if set A do not have two elements with similar image in set B A B a b 2 c 3 d 4 5

61 Function One to One Example. Relation f = {(, w), (2, u), (3, v)} from A = {, 2, 3} to B = {u, v, w, x} is function one to one But relation f = {(, u), (2, u), (3, v)} from A = {, 2, 3} to B = {u, v, w} is not function one to one, because f() = f(2) = u.

62 Function One to One Example. Suppose f : Z Z. Determine whether the functions f(x) = x 2 + and f(x) = x one-to-one? Solution: f(x) = x 2 + is not function one-to-one, f(2) = f(-2) = 5 while 2 2. f(x) = x is not function one-to-one since for a b, a b. Suppose that for x = 2, f(2) = and for x = -2, f(-2) = -3.

63 Function Onto The function f from set A to set B is said to be onto or surjective, if every element of of set B is the image of one or more of the two elements of set A. In other words, all elements of set B are the range of f A B a b 2 c 3 d

64 Function Onto Example. Relation f = {(, u), (2, u), (3, v)} from A = {, 2, 3} to B = {u, v, w} is not function onto since w is not range of f Relation f = {(, w), (2, u), (3, v)} from A = {, 2, 3} to B = {u, v, w} is function onto since all elements of B are range of f.

65 Function Onto Example Suppose f : Z Z. Determine whether f(x) = x 2 + and f(x) = x are function onto? Solution: f(x) = x 2 + is not function onto, since not all of integers are range of f. f(x) = x is funtion onto since for every integer y, there is a real number x, such that x = y +.

66 Function Onto Function one to one but is not onto A a b c B Function onto but is not one to one A a b c d c B 2 3

67 Function Onto It is not function one to one neither onto a b c A B 2 3 d c 4 It is not function A B a b c 2 3 d c 4

68 Function One to One Correspondence Function f is said to be one-toone correspondence or bijection, if it is both one-to-one and onto

69 Function One to One Correspondence Example Function f = {(, u), (2, w), (3, v)} from A = {, 2, 3} to B = {u, v, w} is function one-to-one correspondence since f is one-to-one and onto Function f(x) = x is function one-to-one correspondence since f is one-to-one and onto

70 Invers Function Let function f be one-to-one correspondence from the set A to the set B, we will always be able to find the inverse function of f Inverse function is denoted by f. Suppose that a is an element of set A and b is an element of set B, then f - (b) = a if f(a) = b.

71 Invers Function Example Relation f = {(, u), (2, w), (3, v)} from A = {, 2, 3} to B = {u, v, w} is function one-to-one correspondence. inverse function f is f - = {(u, ), (w, 2), (v, 3)} Thus, f is invertible function

72 Invers Function Example What is the inverse of function f(x) = x. Solution: Function f(x) = x is one-to-one correspondence that we can find its inverse Suppose that f(x) = y, that y = x, then x = y +. Thus, the inverse function is f - (x) = y +.

73 Invers Function Example What is the inverse of function f(x) = x 2 + Solution: Like the previous example, f(x) = x 2 + is not one-to-one correspondence that the inverse does not exist. Thus, f(x) = x 2 + is not invertible.

74 Exercises List the ordered pairs in the relation R from A = {,, 2, 3, 4} to B = {,, 2, 3} where (a,b) R if and only if: a = b a > b a + b = 4 a b

75 Exercises Relations below defined on {, 2, 3, 4}. Determine properties of relations below! {(2, 2), (2, 3), (2, 4), (3, 2), (3, 3), (3, 4)} {(, ), (, 2), (2, ), (2, 2), (3, 3), (4, 4) {(2, 4), (4, 2)} {(, 2), (2, 3), (3, 4)} (, ), (2, 2), (3, 3), (4, 4)

76 Exercises Determine whether the relation R on the set of all people is reflexive, symmetric, antisymmetric, and/or transitive, where (a,b) R if and only if a is taller than b a and b were born on the same day a has the same first name as b

77 Exercises Let R be relation on the set of all URLs (or Web addresses) such that x Ryif and only if Web page at x is the same as the Web page at y. Show that R is an equivalence relation Let R be the relation R on the set of ordered pairs of positive integers such that ((a,b),(c,d)) R if and only if ad=bc. Show that R is an equivalence relation

78 Exercises Which of these are poset? (Z, =) (Z, ) (Z, ) (Z, ) Draw the Hasse for divisibility on the set : {, 2, 3, 4, 5, 6} {3, 5, 7,, 3, 6, 7} {2, 3, 5,, 5, 25} {, 3, 9, 27, 8, 243}

79 References Rosen, Kenneth H., Discrete Mathematics and Its Applications 5th Ed, McGraw-Hill, New York, 23 Munir, Rinaldi., Matematika Diskrit, Edisi Kedua, Penerbit Informatika Bandung, Bandung, 23